# Semantics of Conjectures

Alessio Rolfini
Alma Mater – University of Bologna
Bologna, Italy
<EMAIL_ADDRESS>
###### Abstract

This paper aims to expand and detail the notion of formal semantics of
Conjectures by applying the theoretic-model approach seen in [Hay04], [Pat14]
and related works.

After a short introduction to the concepts and basics of Conjectures, we will
start from the notion of Simple Interpretation of RDF, applying and extending
the semantic rules and conditions to fully cover the concepts and features of
Conjectures.

##  1 Introduction

The semantics of RDF makes it impossible to express contradictory points of
view that act on the same data.

Every approach applied in the past has not solved the main issue, that is,
being able to express different statements whose truth value is not known, or
even in contrast with each other, without fully asserting them, therefore
making them undoubtedly true.

In this paper we introduce and examine the characteristics of conjectures, a
proposed extension to RDF 1.1 that by design allows the expressions of graphs
whose truth value is unknown.

The approach revolves around two main concepts:

  * •

conjecture: a concept whose truth value is not available and its
representation;

  * •

collapse to reality: a mechanism to fully assert, when the time comes, the
truth value of conjectures in their RDF-esque form.

Conjectures encapsulate plain RDF statements or other conjectures in specially
marked named graphs. They can be expressed in a strong form:

    CONJECTURE :deVereWroteHamlet {
        :Hamlet dc:creator :EdwardDeVere .
    }

or, allowing us to be able to include them in plain RDF 1.1 datasets, in a
weak form, where the predicate is expressed as a conjectural predicate:

    @prefix conj0001: <http://example.org/exampleDoc#deVereWroteHamlet>

    GRAPH :deVereWroteHamlet {
      :Hamlet conj0001:creator :EdwardDeVere .
      conj0001:creator conj:isAConjecturalFormOf dc:creator .
    }
    :deVereWroteHamlet prov:wasAttributedTo :JThomasLooney .

Additional statements adding information to the conjecture, but external to
it, are the grounds or ground statements. One of them is the last triple in
the last example.

This paper is organized as follows: in section 2 we show a number of
fundamental concepts and results of the conceptual model of conjectures.

In section 3 we introduce the impact of conjectures onto the standard RDF
Simple interpretation according to [Hay04].

Sections 4 through 7 describe the simple interpretation of the main concepts
of Conjectures: conjectures of individual triples, collapse to reality of
individual triples, conjectures and collapses of graphs, and conjectures
involving blank nodes.

In section 8 we discuss the simple interpretation of the additional features
of conjectures: nested conjectures, conjectural collapses, ad cascading
collapses.

In section 9 we draw some conclusions about the formal model presented here.

##  2 Set-theoretical formal model

Assume the disjoint sets $\mathcal{I}$ (all IRIs), $\mathcal{B}$ (blank
nodes), and $\mathcal{L}$ (literals). An RDF triple is a tuple
$(s,p,o)\in\mathcal{T}=(\mathcal{I}\cup\mathcal{B})\times\mathcal{I}\times(\mathcal{I}\cup\mathcal{B}\cup\mathcal{L})$.

For every RDF predicate $p$, let
$\mathcal{S}_{p}\subseteq(\mathcal{I}\cup\mathcal{B})$ be its domain and
$\mathcal{O}_{p}\subseteq(\mathcal{I}\cup\mathcal{B}\cup\mathcal{L})$ its
range.

We denote with

        x { s p o }

a triple $(s,p,o)$ that is referred to in the examples by the name $x$.

Conjecturing: Conjecturing is the function
$conj:\mathcal{T}\rightarrow\mathcal{T}$ such that, for every RDF triple
$t_{1}=(s_{1},p,o_{1}),conj(t_{1})=(s,p_{s,o},o)$ where:

  1. 1.

Identity of subject: $s_{1}=s$.

  2. 2.

Identity of object: $o_{1}=o$.

  3. 3.

Disjointness: $\forall s_{j},o_{k}$ such that
$(s_{j},p_{s,o},o_{k})\in\mathcal{T}$, we have that $s=s_{j}$ and $o=o_{k}$ .

Conjectures, conjectural predicates, conjecturing triple: given the triple
$(s,p,o)\in\mathcal{T}$, its conjecture is the triple

$conj((s,p,o))=(s,p_{s,o},o).$

Conjectural predicates (or weak predicates) of predicate $p$ are all the
predicates that are members of the set $\mathcal{C}onj_{p}$ , such that:

$\mathcal{C}onj_{p}=\\{cp\in\mathcal{I}|\exists s\in S_{p},o\in
O_{p},conj((s,p,o))=(s,cp,o)\\}$

.

Theorem 1: every conjectural predicate is used in one triple only.

Proof: derives from item 3 of definition 1 (Disjointness): Given two triples
$(s_{1},p_{s,o},o_{1}),(s_{2},p_{s,o},o_{2})\in\mathcal{T}$.

For item 3 of definition 1 (Disjointness), we have that $s_{1}=s_{2}$ and
$o_{1}=o_{2}.\forall(s,p,o)\in\mathcal{T},\exists!p_{s,o}$ such that
$conj((s,p,o))=(s,p_{s,o},o)$.

Corollary 1: the function $conj$ is unique (barring predicate name changes).

Proof: derives immediately from Theorem 1.

Conjectural Form: predicate $q$ is said to be a conjectural form of $p$ if
there exists a pair of subjects and objects $s,o$ such that
$conj((s,p,o))=(s,q,o)$.

Collapsing: Collapsing is a function
$collapses:\mathcal{T}\rightarrow\mathcal{T}$ such that, for every RDF triple
$t=(s,p,o)$, $collapses(s,p,o)=(s,p_{s,o},o)$ iff
$conj((s,p,o))=(s,p_{s,o},o)$, and is undefined otherwise.

Collapsed predicate; collapse: let $(s,p_{s,o},o)$ be a conjecture. The
collapsed predicate of $p_{s,o}$ is the predicate $p$ such that

$collapses(s,p,o)=(s,p_{s,o},o)$

. The collapse is the triple

$((s,p,o),collapse,(s,p_{s,o},o))$

.

##  3 RDF Simple interpretation and Conjectures

In RDF, a simple interpretation $I$ of a vocabulary $V$ consists of:

  1. 1.

A non-empty set $IR$ of resources, called the domain or universe of $I$.

  2. 2.

A set $IP$, called the set of properties of $I$.

  3. 3.

A mapping $IEXT$ from $IP$ into the powerset of $IR\times IR$ i.e. the set of
sets of pairs $<x,y>$ with $x$ and $y$ in $IR$ .

  4. 4.

A mapping $IS$ from IRIs into $IR$ \- in order to map resources and properties

  5. 5.

A partial mapping $IL$ from literals into $IR$ \- in order to map literals

$IEXT(p)$ is the extension of $p$, that is the set of pairs that are the
arguments for which the property $p$ is true.

According to [Pat14], a semantic extension is a set of additional semantic
assumptions that gives IRIs additional meanings on the basis of other
specifications or conventions. When this happens, the semantic extension must
conform to the minimal truth conditions already enunciated, extending from
them to accommodate the additional meanings.

Therefore, we extend the RDF Simple Interpretation adding a new set of
conjectural properties, $IPC$, disjoint from the set of properties $IP$, where
the conjectural predicates are created on the fly.

We add a new mapping $IEXTC$ from $IPC$ to the Cartesian product $IR\times
IR$. $IEXTC(cp)$ identifies the pair for which the property $cp$ is true.

Because of the Disjointness property of the conjecturing function $conj$ seen
in section 1, the pair satisfying the property $cp$ will always be unique.

We need to specify an additional mapping $CONJFORM$ from $IP$ into $IPC$ to
map the conjectural forms of the properties.

Our full simple interpretation of $I$ in RDF with Conjectures is:

  1. 1.

A non-empty set $IR$ of resources, called the domain or universe of $I$.

  2. 2.

A set $IP$, called the set of properties of $I$.

  3. 3.

A set $IPC$, called the set of conjectural properties of $I$. $IPC\cap
IP=\emptyset$

  4. 4.

A mapping $IEXT$ from $IP$ into the powerset of $IR\times IR$ i.e. the set of
sets of pairs $<x,y>$ with $x,y\in IR$ .

  5. 5.

An injective mapping $IEXTC$ from $IPC$ into $IR\times IR$, in other words the
set of pairs $<x,y>$ with $x,y\in IR$. By definition of injective mapping, if
$IEXTC(a)=IEXTC(b)$, then $a=b$, that is, any $cp\in IPC$ uniquely applies to
a specific pair $<x,y>$.

  6. 6.

A mapping $CONJFORM$ from $IP$ into $IPC$ in order to map the conjectural
forms of the properties

  7. 7.

A mapping $IS$ from IRIs into $IR$ \- in order to map resources, properties
and conjectural properties

  8. 8.

A partial mapping $IL$ from literals into $IR$ \- in order to map literals

  9. 9.

A conjecture might be represented as a set of individual statements composing
as a whole the conjecture. This set is a conjectural graph

Figure 1: Schematics of the interpretation

##  4 Conjectures

  * •

if $E$ is a literal then $I(E)=IL(E)$

  * •

if $E$ is an IRI then $I(E)=IS(E)$

  * •

if $E$ is a ground triple $s\>p\>o\>.$ then

    * –

$I(E)=true$ if $I(p)\in IP$ and

    * –

the pair $<I(s),I(o)>\in IEXT(I(p))$

    * –

otherwise $I(E)=false$.

  * •

if $E$ is a Conjecture triple $s\>cp\>o\>.$ then

    * –

$I(E)=true$ if $I(cp)\in IPC$ and

    * –

the pair $<I(s),I(o)>\in IEXTC(I(cp))$

    * –

$I(cp)\in CONJFORM(I(p))$ for some $I(p)\in IP$

    * –

otherwise $I(E)=false$.

  * •

if $E$ is a ground RDF graph then $I(E)=false$ if $I(E^{\prime})=false$ for
some triple $E^{\prime}\in E$, otherwise $I(E)=true$.

  * •

if $E$ is a ground conjectural graph then $I(E)=false$ if
$I(E^{\prime})=false$ for some triple $E^{\prime}\in E$, otherwise
$I(E)=true$.

The last clause captures the definition of conjectural graph seen before.

###  4.1 Model

Given a RDF graph $G$, we say that the interpretation $I$ is a model of the
graph $G$ if all the triples of graph $G$ are satisfied, that is, if they are
true according to rules of $I$. In this case, we can say that Interpretation
$I$ satisfies $G$.

The notion of model is at the basis of the (simple) entailment: following
standard terminology, we say that $I$ (simply) satisfies $G$ when $I(G)=true$,
that $G$ is (simply) satisfiable when a simple interpretation exists which
satisfies it, otherwise (simply) unsatisfiable, and that a graph $E$ simply
entails a graph $G$ when every interpretation which satisfies $E$ also
satisfies $G$. If two graphs $G$ and F each entail the other then they are
logically equivalent. [Pat14]

Is our interpretation $I$ a model of this conjecture graph?

    :deVereWroteHamlet {
       :Hamlet conj001:creator :EdwardDeVere .
       conj001:creator conj:isAConjecturalFormOf dc:creator .
    }

  * •

$IR=\\{dVWH,h,c,cc1,e,iacf\\}$

  * •

$IP=\\{c,iacf\\}$

  * •

$IPC=\\{cc1\\}$

The functions:

  * •

$IS(:deVereWroteHamlet)\rightarrow dVWH$

  * •

$IS(:Hamlet)\rightarrow h$

  * •

$IS(conj001:creator)\rightarrow cc1$

  * •

$IS(:EdwardDeVere)\rightarrow e$

  * •

$IS(conj:isAConjecturalFormOf)\rightarrow iacf$

  * •

$IS(dc:creator)\rightarrow c$

  * •

$IEXT(c)=\emptyset$

  * •

$IEXT(iacf)=\\{<cc1,c>\\}$

  * •

$IEXTC(cc1)=\\{<h,e>\\}$

  * •

$CONJFORM(c)=cc1$

IEXT(c) is the empty set. We are still dealing with a conjecture, there is no
assertion regarding any ”real” property (in this case $dc:creator$) yet.

All the clauses seem to hold. We can say that simple interpretation
(s-interpretation) $I$ is a model of our graph.

##  5 Collapse to reality

At a certain point, someone may want to consider our conjectures as true.

In this case we would need to specify a mechanism in order to express the
statements in the conjecture in full force.

This mechanism is called ”collapse to reality”.

It consists of exactly two triples added to the base, where:

  * •

in the first new triple, the ”effective” property is used instead of the
corresponding conjectural property.

  * •

the first new triple is then declared to ”collapse” the conjecture .

Two new triples are added, nothing gets replaced or deleted. So we can keep
track of what’s happening and all the relationships between the graphs.

Let’s reason on an example of a collapse:

    :attr1 {
        :Hamlet conj003:creator :Shakespeare .
        conj003:creator conj:isAConjecturalFormOf dc:creator.
    }

    :attr1Cot {
        :Hamlet dc:creator :Shakespeare .
    }

    :attr1Cot  conj:collapses :attr1 .

$:attr1$ is collapsed by adding a triple $:attr1Cot$ where the conjectural
predicate is replaced by the corresponding ”real” predicate.

The final additional triple functions as the explication of the collapse.
$:attr1Cot$ is declarad to collapse the conjecture $:attr1$ by means of the
property ”conj:collapses”.

For the sake of clarity:

  * •

the conjecture triple to be collapsed will be the ”conjecture triple”;

  * •

the new triple collapsing the conjecture will be the ”collapsing triple”;

  * •

the last triple will be the ”collapse triple”.

###  5.1 Interpretation $[I+COLLAPSE]$

We need to extend once more our Interpretation with Conjectures $I$ by adding
the Collapse to Reality rules.

We define a new mapping $collapses$ from triples to triples that maps the
relationship between the conjecture triple and its collapsing triple:

$collapses(s,p,o)=(s,cp,o)$ iff $conj(s,p,o)=(s,cp,o)$.

The collapse collapses the conjecture triple if, and only if, the latter is
the (unique) conjecture of the collapsing triple.

We can clearly see that the ”collapses triple” is just the translation into
RDF of the definition of collapse we have introduced in section 1, that is,
the triple:

$((s,p,o),\quad collapses,\quad(s,cp,o))$

From a more formal point of view, given a conjectural triple $E$, we add the
triple $E_{cot}$ as a collapsing triple $\\{s\quad p\quad o.\\}$.

The semantic conditions of the collapse to reality should be the following:

  * •

let $E$ be a conjecture triple $\\\ \\{s\quad cp\quad o\quad.\\}$

let $E_{cot}$ be a collapsing triple $\\\ \\{s\quad p\quad o\quad.\\}$

and finally let $E_{collapse}$ be the collapse triple $\\\
\\{E_{cot}\quad\mathit{collapses}\quad E\\}$

then

    * –

$I(E)=true$ if $I(cp)\in IPC$ and

    * –

$I(E_{cot})=true$ if $I(p)\in IP$ and

    * –

$CONJFORM(I(p))=I(cp)$ and

    * –

the pair $<I(s),I(o)>\in IEXTC(I(cp))$ and

    * –

the pair $<I(s),I(o)>\in IEXT(I(p))$ and

    * –

$I(E_{collapse})=true$ if $collapses(I(E_{cot}))=I(E)$, that is:
$collapses(I(s),I(p),I(o))=(I(s),I(cp),I(o))$

    * –

otherwise $I(E)=false$ and $I(E_{cot})=false$ and $I(E_{collapse})=false$.

###  5.2 Model

Now, let’s check if our Interpretation $[I+COLLAPSE]$ can be a model of our
example graph.

  * •

$IR=\\{a1,h,cc3,s,iacf,c,a1cot,cl\\}$

  * •

$IP=\\{iacf,c,cl\\}$

  * •

$IPC=\\{cc3\\}$

The functions:

  * •

$IS(:attr1)\rightarrow a1$

  * •

$IS(:Hamlet)\rightarrow h$

  * •

$IS(conj003:creator)\rightarrow cc3$

  * •

$IS(:Shakespeare)\rightarrow s$

  * •

$IS(conj:isAConjecturalFormOf)\rightarrow iacf$

  * •

$IS(dc:creator)\rightarrow c$

  * •

$IS(:attr1Cot)\rightarrow a1cot$

  * •

$IS(conj:collapses)\rightarrow cl$

  * •

$IEXT(iacf)=\\{<cc3,c>\\}$

  * •

$IEXT(c)=\\{<h,s>\\}$

  * •

$IEXT(cl)=\\{<a1,a1cot>\\}$

  * •

$IEXTC(cc3)=\\{<h,s>\\}$

  * •

$CONJFORM(c)=cc3$

Let’s check our semantic conditions for the collapse to reality:

  * •

$cc3\in IPC$? Yes;

  * •

$c\in IP$? Yes;

  * •

$CONJFORM(c)=cc3$? Yes;

  * •

The pair $<h,s>\in IEXTC(cc3)$? Yes;

  * •

The pair $<h,s>\in IEXTC(c)$? Yes;

  * •

$collapses(h,c,s)=(h,cc3,s)$? Yes, because $CONJFORM(c)=cc3$, hence
$conj(h,c,s)=(h,cc3,s)$

All of them seem to hold.

Hence, our interpretation $[I+COLLAPSE]$ is a model of our graph.

As we will see in section 6.2, the collapsing triple and the collapse triple
can be safely merged into a collapse graph:

    :attr1 {
        :Hamlet conj003:creator :Shakespeare .
        conj003:creator conj:isAConjecturalFormOf dc:creator.
    }

    :collapseOfattr1 {
        :Hamlet dc:creator :Shakespeare .
        :collapseOfattr1 conj:collapses :attr1 .
    }

##  6 Conjectural Graphs and Collapse Graphs

###  6.1 Conjectural Graphs

A conjectural graph is a representation of a conjecture by means of a set of
individual statements composing as a whole the said conjecture.

A ground Conjectural Graph is a graph with no blank nodes.

All triples inside a Conjectural Graph must be either with a conjectural
predicate used exactly once, or a ”conj:isAConjecturalFormOf” triple
connecting a conjectural predicate to an original one.

The semantic conditions for Conjectural Graphs are:

  * •

if $E$ is a literal then $I(E)=IL(E)$

  * •

if $E$ is an IRI then $I(E)=IS(E)$

  * •

if $E$ is a Conjecture triple $s\>cp\>o\>.$ then

    * –

$I(E)=true$ if $I(cp)\in IPC$ and

    * –

the pair $<I(s),I(o)>\in IEXTC(I(cp))$

    * –

$I(cp)\in CONJFORM(I(p))$ for some $I(p)\in IP$ and

    * –

$E$ must be unique in the graph

    * –

otherwise $I(E)=false$.

  * •

if $E$ is a triple $cpconj:isAConjecturalFormOfp$ then

    * –

$I(E)=true$ if $I(cp)\in IPC$ and

    * –

$I(p)\in IP$ and

    * –

$I(cp)\in CONJFORM(I(p))$ and

    * –

$I(conj:isAConjecturalFormOf)\in IP$

    * –

the pair $<I(cp),I(p)>\in IEXT(I(conj:isAConjecturalFormOf))$

    * –

otherwise $I(E)=false$

  * •

if $E$ is a ground conjectural graph then $I(E)=false$ if
$I(E^{\prime})=false$ for some triple $E^{\prime}\in E$, otherwise
$I(E)=true$.

###  6.2 Collapse Graphs

Collapse graphs are graphs that connect explicitly with a conjecture and make
it possible to track which conjecture was collapsed.

They must be composed at least of:

  * •

the ”effective form” of the conjecture to be collapsed

  * •

a ”conj:collapse” triple connecting the conjecture to the collapse

The semantic conditions are:

  * •

let $E_{conj}$ be a conjecture triple $s\>cp\>o\>.$

  * •

let $E_{collapse}$ be a collapse graph

  * •

if $E$ is a literal then $I(E)=IL(E)$

  * •

if $E$ is an IRI then $I(E)=IS(E)$

  * •

if $E$ is a triple $s\>p\>o\>.$ then

    * –

$I(E)=true$ if $I(p)\in IP$ and

    * –

the pair $<I(s),I(o)>\in IEXT(I(p))$

    * –

$CONJFORM(I(p))=I(cp)$ where $cp$ is the conjectural predicate of the
conjecture to be collapsed

    * –

otherwise $I(E)=false$.

  * •

if $E$ is a triple $E_{collapse}\quad conj:collapses\quad E_{conj}$ then

    * –

$I(E)=true$ if $I(conj:collapses)\in IP$ and

    * –

the pair $<I(E_{collapse}),I(E_{conj})>\in IEXT(I(conj:collapses))$

    * –

otherwise $I(E)=false$

  * •

if $E$ is a ground conjectural graph then $I(E)=false$ if
$I(E^{\prime})=false$ for some triple $E^{\prime}\in E$, otherwise
$I(E)=true$.

##  7 Blank Nodes

With conjectures, we would want to be able to express even more ”uncertain”
concepts involving unnamed entities or unspecified values.

In RDF this is done through blank nodes, which indicate the existence of an
entity without using a IRI to identify any particular one.

In this section we will use a simple sentence implying the reliance on a blank
node, namely ”Muammar al-Qaddafi claimed that the author of Othello was
someone who was an Arab” 111He really did it in December 1988 - see ”William
Shakespeare’s Othello” by Jibesh Bhattacharyya, ISBN 9788126904747.

In this case we don’t know who someone is, and we can’t identify it with any
IRIs.

Nevertheless, the information we are conveying with our conjecture maintains
some degree of meaningfulness, and it definitely is something we could reason
upon.

Sticking to our examples’ style, we could say it like this: ”it is conjectured
that Othello was written by somebody and this somebody was an Arab. And this
claim has been attributed to Muammar al-Qaddafi”.

In RDF:

    :ArabWroteOthello {
      :Othello conj002:creator _:z .
      conj002:creator conj:isAConjecturalFormOf dc:creator .
      _:z dbpedia-owl:nationality :Arab .}

    :ArabWroteOthello prov:wasAttributedTo :MalQaddafi .

It comes pretty natural to conceive the term ”somebody” as a blank node.

###  7.1 Interpretation $[I+A]$

In order to deal with blank nodes, we need to add a new mapping $A$ from blank
nodes into IR.

Therefore, we extend our interpretation $I$:

  * •

$[I+A](x)=I(x)$ for names

  * •

$[I+A](x)=A(x)$ if x is a blank node.

We add a couple of semantic conditions to our interpretation for blank nodes,
one is the ”standard” one for RDF graphs, the other one is for conjectures:

  * •

if $E$ is a RDF graph then $I(E)=true$ if $[I+A](E)=true$ for some mapping $A$
from the set of blank nodes in $E$ to $IR$, otherwise $I(E)=false$.

  * •

if $E$ is a conjectural graph then $I(E)=true$ if $[I+A](E)=true$ for some
mapping $A$ from the set of blank nodes in $E$ to $IR$, otherwise
$I(E)=false$.

###  7.2 Model

Is our Interpretation $[I+A]$ with blank nodes a model of our example graph?

Let’s reason on $:ArabWroteOthello$ part only.

Be $A$ our blank nodes mapping into $IR$: $\\_:z\rightarrow zz$.

Our interpretation $[I+A]$ will be:

  * •

$IR=\\{awo,o,c,cc2,iacf,n,a,zz\\}$

  * •

$IP=\\{c,iacf,n\\}$

  * •

$IPC=\\{cc2\\}$

The functions:

  * •

$IS(:ArabWroteOthello)\rightarrow awo$

  * •

$IS(:Othello)\rightarrow o$

  * •

$IS(conj002:creator)\rightarrow cc2$

  * •

$IS(dc:creator)\rightarrow c$

  * •

$IS(dbpedia-owl:nationality)\rightarrow n$

  * •

$IS(:Arab)\rightarrow a$

  * •

$IS(conj:isAConjecturalFormOf)\rightarrow iacf$

  * •

$IEXT(c)=\emptyset$

  * •

$IEXT(iacf)=\\{<cc2,c>\\}$

  * •

$CONJFORM(c)=cc2$

  * •

$IEXT(n)=\\{<zz,a>\\}$

  * •

$IEXTC(cc2)=\\{<o,zz>\\}$

Even in this case $IEXT(c)$ is empty because there is no assertion regarding
the property $dc:creator$ yet, so our conjecture is, well, still a conjecture.

The last two functions hold because of the mapping $A$ of our blank node.

All the clauses are true. Our interpretation $I+A$ is a model of our graph.

This approach allows us to reason with the blank nodes in what-if scenarios,
where we could define the $A$ mapping from blank nodes to specific resources
of our choice, therefore exploring new relationships arising between the
triples.

##  8 Further applications of conjectures

###  8.1 Conjecture of a conjecture - nested conjectures

Let’s imagine we want to express a conjecture on another conjecture. Of
course, the process can involve as many conjectures over conjectures we want.

Such as:

    :conjecture01 {
        :Hamlet conj004:creator :EdwardDeVere .
        conj004:creator conj:isAConjecturalFormOf dc:creator .
    }

    :conjecture02 {
     :conjecture01 conj004:wasAttributedTo :JThomasLooney .
     conj004:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .
    }

    :conjecture03 {
    :conjecture02 conj004:wasInformedBy <https://bit.ly/3wSH76A> .
    conj004:wasInformedBy conj:isAConjecturalFormOf prov:wasInformedBy .
    }

    :conjecture03 prov:wasAttributedTo :FabioVitali .

They are three different conjectures, one becoming the subject of the other
one:

  * •

the first one says that Hamlet is conjectured to have been written by Edward
De Vere;

  * •

the second one says that the previous conjecture could possibly have been made
by J. Thomas Looney;

  * •

The third one says that it might be that the second conjecture could have been
brought to light by the resource with URL https://bit.ly/3wSH76A.

The fourth triple says that the last conjecture has been attributed to Fabio
Vitali.

Reading it (more or less) backwards: Fabio Vitali has stated that the resource
at https://bit.ly/3wSH76A might have brought to light that J. Thomas Looney
could have possibly said that Hamlet is conjectured to have been written by
Edward De Vere.

We could imagine it as a stack, or a stair, where at level 0 we have the first
conjecture, and then as we get on the higher steps, the conjectures we find
are built with the conjecture at the level below as a subject (or object, or
both):

    Level 2:            E{3}={E{2}, cp3, o3}
    Level 1:                  E{2}=(E{1}, cp2, o2}
    Level 0:                        E{1}=(s1, cp1, o1)

Developing the Level 2 we see they are nested in each other:
$E{3}=(E{1},cp2,o2),cp3,o3)=(((s1,cp1,o1),cp2,o2),cp3,o3)$

####  8.1.1 Interpretation $[I+NESTEDCONJ]$

In order to delineate the rules for the conjectures of conjectures (or nested
conjectures), we can reason with pairs of conjectures.

As stated before, the conjectures at levels $>0$ could be based on lower-level
conjectures as their subject or object, or both.

For the sake of conciseness, we momentarily limit ourselves to reason with the
case of the lower-level conjectures as their subject only.

  1. 1.

if $E_{0}$ is a ground conjecture $(s_{0},cp_{0},o_{0})$,
$E_{1}=(E_{0},cp_{1},o_{1})$

  2. 2.

if $E_{1}$ is a ground conjecture $(E_{0},cp_{1},o_{1})$,
$E_{2}=(E{1},cp_{2},o_{2})$ […]

  3. 3.

if $E_{k-1}$ is a ”higher level” ground conjecture
$(E_{k-2},cp_{k-1},o_{k-1})$, $E_{k}=(E_{k-1},cp_{k},o_{k})$

  4. 4.

if $E_{k}$ is a ”higher level” ground conjecture $(E_{k-1},cp_{k},o_{k})$,
$E_{k+1}=(E_{k},cp_{k+1},o_{k+1})$

We should have the following cases:

  * •

base case:

    * –

$E_{0}=(s_{0},p_{0},o_{0})$

  * •

1st cases:

    * –

$E_{1}=(E_{0},cp_{1},o_{1})$

    * –

$E_{1}=(s_{1},cp_{1},E_{0})$

  * •

$k^{th}$ cases:

    * –

$E_{k}=(E_{k-1)},cp_{k},o_{k})$

    * –

$E_{k}=(s_{k},cp_{k},E_{k-1})$

Let’s extend our interpretation $I$ adding new rules.

The extension will be subdivided into cases, depending on the type of
conjectures to be evaluated.

We must also enforce an order on the operations: we start from the
conjecture(s) at the ”lowest level”, that is, the one(s) not involving other
conjectures, evaluate them and ”climb” up the ”stair”.

Therefore, the extension to the interpretation $I$ will be:
Base case \- for the lowest level conjecture at level 0:

  * •

let $E_{0}$ be a conjecture triple $\\\ \\{s_{0}\quad cp_{0}\quad
o_{0}\quad.\\}$ then

    * –

$I(E_{0})=true$ if $I(cp_{0})\in IPC$ and

    * –

the pair $<I(s_{0}),I(o_{0})>\in IEXTC(I(cp_{0}))$

    * –

otherwise $I(E_{0})=false$

As we ”climb” up the ”stair” and get to the conjecture $E_{k}$, we can assume
that the conjecture $E_{k-1}$ has already been proved by the previous steps.

$K^{th}$ Case S \- for conjectures at the first step and above having another
conjecture as the subject:

  * •

let $E_{k-1}$ be a conjecture triple

let $E_{k}$ be a conjecture triple $\\\ \\{E_{k-1}\quad cp_{k}\quad
o_{k}\quad.\\}$

then

    * –

$I(E_{k})=true$ if $I(E_{k-1})=true$ and

    * –

$I(cp_{k})\in IPC$ and

    * –

the pair $<I(E_{k-1}),I(o_{k})>\in IEXTC(I(cp_{k}))$

    * –

otherwise $I(E_{k})=false$

$K^{th}$ Case O \- for conjectures at the first step and above having another
conjecture as the object:

  * •

let $E_{k-1}$ be a conjecture triple

let $E_{k}$ be a conjecture triple $\\\ \\{s_{k}\quad cp_{k}\quad
E_{k-1}\quad.\\}$

then

    * –

$I(E_{k})=true$ if $I(E_{k-1})=true$ and

    * –

$I(cp_{k})\in IPC$ and

    * –

the pair $<I(s_{k}),I(E_{k-1})>\in IEXTC(I(cp_{k}))$

    * –

otherwise $I(E_{k})=false$

$K^{th}$ Case SO \- for conjectures at the first step and above having another
conjecture as the subject and yet another one as the object:

  * •

let $E_{(k-1)a}$ be a conjecture triple

let $E_{(k-1)b}$ be a conjecture triple

let $E_{k}$ be a conjecture triple $\\\ \\{E_{(k-1)a}\quad cp_{k}\quad
E_{(k-1)b}\quad.\\}$

then

    * –

$I(E_{k})=true$ if $I(E_{(k-1)a})=true$ and

    * –

$I(E_{(k-1)b})=true$ and

    * –

$I(cp_{k})\in IPC$ and

    * –

the pair $<I(E_{(k-1)a}),I(E_{(k-1)b})>\in IEXTC(I(cp_{k}))$

    * –

otherwise $I(E_{k})=false$

####  8.1.2 Model

Is our Interpretation $[I+NESTEDCONJ]$ a model of the nested conjectures
example seen before?

We define the sets:

  * •

$IR=\\{c1,h,cc4,edv,iacf,c,c2,cwa4,jtl,pwa,c3,cwib4,http,pwib,fv\\}$

  * •

$IP=\\{c,iacf,pwa,pwib\\}$

  * •

$IPC=\\{cc4,cwa4,cwib4\\}$

The functions:

  * •

$IS(:conjecture01)\rightarrow c1$

  * •

$IS(:Hamlet)\rightarrow h$

  * •

$IS(conj004:creator)\rightarrow cc4$

  * •

$IS(:EdwardDeVere)\rightarrow edv$

  * •

$IS(conj:isAConjecturalFormOf)\rightarrow iacf$

  * •

$IS(dc:creator)\rightarrow c$

  * •

$IS(:conjecture02)\rightarrow c2$

  * •

$IS(conj004:wasAttributedTo)\rightarrow cwa4$

  * •

$IS(:JThomasLooney)\rightarrow jtl$

  * •

$IS(prov:wasAttributedTo)\rightarrow pwa$

  * •

$IS(:conjecture03)\rightarrow c3$

  * •

$IS(conj004:wasInformedBy)\rightarrow cwib4$

  * •

$IL(<https://bit.ly/3wSH76A>)\rightarrow http$

  * •

$IS(prov:wasInformedBy)\rightarrow pwib$

  * •

$IS(:FabioVitali)\rightarrow fv$

  * •

$IEXT(iacf)=\\{<cc4,c>,<cwa4,pwa>,<cwib4,pwib>\\}$

  * •

$IEXT(c)=\emptyset$

  * •

$IEXT(pwa)=\\{<c3,fv>\\}$

  * •

$IEXT(pwib)=\emptyset$

  * •

$IEXTC(cc4)=\\{<h,ev>\\}$

  * •

$IEXTC(cwa4)=\\{<c1,jtl>\\}$

  * •

$IEXTC(cwib4)=\\{<c2,http>\\}$

  * •

$CONJFORM(c)=cc4$

  * •

$CONJFORM(cwa4)=pwa$

  * •

$CONJFORM(cwib4)=pwib$

Let’s check the validity of the rules of the new semantic extension.

We must start from the conjecture not depending on other conjecture, that is
$:conjecture01$, and we use the Base Case:

  * •

Is $cc4\in IPC$? Yes

  * •

Is the pair $<h,ev>\in IEXTC(cc4)$ Yes

The ”base” conjecture seems to hold. We can say that $c1=true$.

Now we ”climb the stair” to $:conjecture02$. Since it is based on another
conjecture ($:conjecture01$, already proved true) as its subject, we use
”$k^{th}$ Case S”

  * •

is $c1=true$? Yes

  * •

is $cwa4\in IPC$? Yes

  * •

is the pair $<c1,jtl>\in IEXTC(cwa4)$? Yes

Then we can say that $c2=true$.

Let’s climb one step higher. $:conjecture03$ is based on $:conjecture02$ as
its subject. We still use ”$k^{th}$ Case S”

  * •

is $c2=true$? Yes

  * •

is $cwib4\in IPC$? Yes

  * •

is the pair $<c2,http>\in IEXTC(cwib4)$? Yes

$\rightarrow c3=true$.
The last triple $:conjecture03prov:wasAttributedTo:FabioVitali$ is satisfied
by the rules of the simple intepretation $I$
Everything seems to hold.

We can say that our Interpretation $[I+NESTEDCONJ]$ satisfies all the triples
of the graph.

###  8.2 Conjecture of a collapse

Let us consider the following sentence:

> According to Encyclopaedia Britannica (https://bit.ly/3qgakFT), Samuel
> Johnson attributed Hamlet to William Shakespeare, and he was right in saying
> so.

This sentence is more than a mere collapse: it is a conjecture by an article
in Encyclopedia Britannica a) attributing to Samuel Johnson a conjecture
(about the authorship of Hamlet), and b) expressing total confidence in such
conjecture (i.e., collapsing it).

Its representation is:

    :attribution01 {
      :Hamlet conj005:creator :WilliamShakespeare .
      conj005:creator conj:isAConjecturalFormOf dc:creator .
    }

    :collapseOfAttribution01 {
      :attribution01 conj005:wasAttributedTo :SamuelJohnson .
      conj005:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .

      :collapseOfAttribution01 conj005:collapses :attribution01 .
      conj005:collapses conj:isAConjecturalFormOf conj:collapses .
    }

    :collapseOfattribution01 prov:wasInformedBy <https://bit.ly/3qgakFT> .

In this case we have a conjecture of a collapse that, if collapsing, as a side
effect, triggers the collapse of another conjecture, therefore asserting it in
full force. Please note that, in its current and ”uncollapsed” status,
everything is conjectured.

We can see the familiar predicate $collapses$ in its conjectural form, not yet
in force in this case. This predicate, once met in its ”effective” form, will
trigger the collapse of its subject. We will see it in the next section.

###  8.3 Cascading collapses

Cascading collapses are all the collapses that take place recursively when
multiple and nested conjectures of collapses collapse.

For example, what if we wanted to state that the $collapseOfAttribution01$ is
true? We would collapse it by adding the ”collapsing triple” and the ”collapse
triple” as per the rules of Interpretation [I + COLLAPSE] seen before in 5.1.

For the sake of simplicity, we resort to the notion of collapse graph (6.2).
According to that, we can safely group the added triples into the collapse
graph $:collapseOfcollapseOfAttribution01$.

Now that our conjecture of a collapse is declared true, therefore the collapse
is true and effective, we have to enforce it, collapsing $:attribution01$

Generalising, once a conjecture of a collapse is declared true, we must
enforce a new special rule for its collapse to reality:

Whenever $collapses$ is met in its effective form inside a collapse graph, its
object, if it is a conjecture, must be collapsed.

If the object is a conjecture triple $s\>cp\>o$, we need to add the collapsing
triple and the collapse triple.

Getting back to our example, we can see that the collapse triple is already
defined in its effective form, in fact we have that:

        :collapseOfAttribution01 conj:collapses :attribution01 .

So we just need to add the collapsing triple, stating the conjecture in its
full force, in the graph where the $conj:collapses$ property is met in full
force.

The result will be as follows:

    :attribution01 {
      :Hamlet conj005:creator :WilliamShakespeare .
      conj005:creator conj:isAConjecturalFormOf dc:creator .
    }

    :collapseOfAttribution01 {
      :attribution01 conj005:wasAttributedTo :SamuelJohnson .
      conj005:wasAttributedTo conj:isAConjecturalFormOf prov:wasAttributedTo .

      :collapseOfAttribution01 conj005:collapses :attribution01  .
      conj005:collapses conj:isAConjecturalFormOf conj:collapses .
    }
    :collapseOfattribution01 prov:wasInformedBy <https://bit.ly/3qgakFT> .

    :collapseOfcollapseOfAttribution01  {
        :attribution01 prov:wasAttributedTo :SamuelJohnson .

        :collapseOfAttribution01 conj:collapses :attribution01 .

        :collapseOfcollapseOfAttribution01 conj:collapses :collapseOfAttribution01 .
           .

        :Hamlet dc:creator :WilliamShakespeare.
    }

A collapse graph is built for the first collapse of the conjecture of the
collapse, and then, at the end, the final ”effective” triple.

We need to extend the Interpretation [I + COLLAPSE] seen before in 5.1.

After we collapse the conjecture of the collapse following the rules in
[I+COLLAPSE], we will find the property $collapses$ in its effective form
inside the collapse graph.

We need to extend the notion of $collapses$ defining it from collapse graphs
to conjectural graphs: $collapses(G)=(E)$ iff $\forall(s,cp,o)\in
E,conj(s,p,o)=(s,cp,o)with(s,p,o)\in G$.

If the object of the $collapses$ property is a triple, which can be considered
as a special case of a graph, we will add its effective form to the
conjectural graph $collapses$ is in.

If the object of $collapses$ property is a conjectural graph, we need to add
the effective forms of all its conjectures.

One important thing to be aware of is that we disregard the subject of the
$collapses$ property: the effective form of the conjecture(s) in the object(s)
is added to the current collapse graph no matter their subject(s).

In the semantic conditions below we will simplify for clarity by adopting the
notion of a generic graph as the theoretical subject of a $collapses$
property. The graphs will in fact be more than one, if the subjects of the
conjectures which are objects of $collapses$ are distinct (as in the example
above). Also The conjecture graphs/triples can be more than one.

The semantic conditions are:

conjecture triple:

  * •

let $E$ be a conjecture triple $\\\ \\{s\quad cp\quad o\quad.\\}$

let $E_{g}$ be a generic graph

let $E_{cg}$ be a collapse graph $\\\
\\{E_{g}\quad\mathit{conj:collapses}\quad E\\\ s\quad p\quad o.\\}$

then

    * –

$I(E)=true$ if $I(cp)\in IPC$ and

    * –

$I(E_{cg})=true$ if $I(p)\in IP$ and

    * –

$CONJFORM(I(p))=I(cp)$ and

    * –

the pair $<I(s),I(o)>\in IEXTC(I(cp))$ and

    * –

the pair $<I(s),I(o)>\in IEXT(I(p))$ and

    * –

$I(conj:collapses)\in IP$ and

    * –

the pair $<I(E_{g}),I(E>\in IEXT(I(conj:collapses))$

    * –

otherwise $I(E)=false$ and $I(E_{cg})$ is false.

conjectural graph: If we are dealing with a conjectural graph as the object of
$conj:collapses$:

  * •

let $E$ be a conjectural graph $\\\ \\{s_{0}\quad cp_{0}\quad o_{0}\quad.\\\
{[...]}\\\ s_{k}\quad cp_{k}\quad o_{k}\quad.\\}$

let $E_{g}$ be a generic graph

let $E_{cg}$ be a collapse graph $\\\ \\{s_{0}\quad p_{0}\quad o_{0}.\\\
{[...]}\\\ s_{k}\quad p_{k}\quad o_{k}\quad.\\\
E_{g}\quad\mathit{conj:collapses}\quad E\\}$

then

    * –

$I(E)=true$ if $I(cp_{k})\in IPC\forall cp_{k}inE$ and

    * –

$I(E_{cg})=true$ if $I(p_{k})\in IP\forall p_{k}inE_{cg}$ and

    * –

$CONJFORM(I(p_{k}))=I(cp_{k})\forall p_{k}inE$ and

    * –

any pair $<I(s_{k}),I(o_{k})>\in IEXTC(I(cp_{k}))$ and

    * –

any pair $<I(s_{k}),I(o_{k})>\in IEXT(I(p_{k}))$ and

    * –

$I(conj:collapses)\in IP$ and

    * –

the pair $<I(E_{g}),I(E)>\in IEXT(I(conj:collapses))$

    * –

otherwise $I(E)=false$ and $I(E_{cg})$ is false.

####  8.3.1 Model

Let’s check once again if the interpretation can be a model for our last
example.

We define the sets:

  * •

$IR=\\{a1,h,cc5,ws,iacf,c,coa1,c5wat,sj,pwat,c5cl,cl,pwib,url,ccoa1\\}$

  * •

$IP=\\{c,iacf,pwat,pwib,cl\\}$

  * •

$IPC=\\{cc5,c5wat,c5cl\\}$

The functions:

  * •

$IS(:attribution01)\rightarrow a1$

  * •

$IS(:Hamlet)\rightarrow h$

  * •

$IS(conj005:creator)\rightarrow cc5$

  * •

$IS(:WilliamShakespeare)\rightarrow ws$

  * •

$IS(conj:isAConjecturalFormOf)\rightarrow iacf$

  * •

$IS(dc:creator)\rightarrow c$

  * •

$IS(:collapseOfAttribution01)\rightarrow coa1$

  * •

$IS(conj005:wasAttributedTo)\rightarrow c5wat$

  * •

$IS(:SamuelJohnson)\rightarrow sj$

  * •

$IS(prov:wasAttributedTo)\rightarrow pwat$

  * •

$IS(conj005:collapses)\rightarrow c5cl$

  * •

$IS(conj:collapses)\rightarrow cl$

  * •

$IS(prov:wasInformedBy)\rightarrow pwib$

  * •

$IL(<https://bit.ly/3qgakFT>)\rightarrow url$

  * •

$IS(:collapseOfcollapseOfAttribution01)\rightarrow ccoa1$

  * •

$IEXT(c)=\\{<h,ws>\\}$

  * •

$IEXT(iacf)=\\{<cc5,c>,<c5wat,pwat>,<c5cl,cl>\\}$

  * •

$IEXT(pwat)=\\{<a1,sj>\\}$

  * •

$IEXT(cl)=\\{<a1,coa1>,<coa1,ccoa1>\\}$

  * •

$IEXT(pwib)=\\{<coa1,url>\\}$

  * •

$IEXTC(cc5)=\\{<h,ws>\\}$

  * •

$IEXTC(c5wat)=\\{<a1,sj>\\}$

  * •

$IEXTC(c5cl)=\\{<a1,coa1>\\}$

  * •

$CONJFORM(c)=cc5$

  * •

$CONJFORM(pwat)=c5wat$

  * •

$CONJFORM(cl)=c5cl$

Let’s check now the rules. We focus the check on the cascading collapses only.

We need to proceed in steps. The first $conj:collapses$ in our final collapse
graph has $:attribution01$ as its object. We can consider $:attribution01$ as
a single triple, as it contains only one conjecture:

  * •

is $cc5\in IPC$? Yes;

  * •

is $c\in IP$? Yes;

  * •

is $CONJFORM(c)=cc5$? Yes;

  * •

is the pair $<h,ws>\in IEXTC(cc5)$? Yes;

  * •

is the pair $<h,ws>\in IEXT(c)$? Yes;

  * •

is $cl\in IP$? Yes;

  * •

the pair $<a1,coa1>\in IEXT(cl)$?Yes.

It seems to hold.

Let’s get to the second $con:collapses$ of our final collapse graph. It has
$:collapseOfAttribution01$ as its object. It is a graph. Let’s apply rules for
the graphs:

  * •

$c5wat,c5cl\in IPC$? Yes;

  * •

$pwat,cl\in IP$? Yes;

  * •

$CONJFORM(c5wat)=pwat;CONJFORM(c5cl)=cl?$ Yes;

  * •

$<a1,sj>\in IEXTC(c5wat)$? Yes; $<a1,coa1>\in IEXTC(c5cl)$? Yes;

  * •

$<a1,sj)>\in IEXT(pwat)$? Yes; $<a1,coa1>\in IEXT(cl)$? Yes;

  * •

$cl\in IP$? Yes;

  * •

the pair $<a1,coa1)>\in IEXT(cl)$? Yes

It can be easily verified that all the other triples of the example can be
satisfied by the interpretation [I + COLLAPSE].

Our interpretation is a model of the graph.

##  9 Conclusions

In this paper we have attempted to build a formal representation of the
semantics of the conjectures.

We have started with the formal model and extended the simple interpretation
and model of RDF 1.1.

Then we have explored the key features of the conjectures and verified that
they fit in the semantics, expanding it when needed.

The conjectures allow to express concepts whose truth value is unknown,
without asserting it, something that is currently missing in RDF.

With this work we have demonstrated that, with a somewhat limited extension to
its model, it is possibile to add this powerful feature to RDF 1.1.

## References

  * [Hay04] Patrick Hayes  “RDF Semantics”, 2004  URL: https://www.w3.org/TR/rdf-mt/
  * [Pat14] Peter F.-Schneider Patrick J.  “RDF 1.1 Semantics”, 2014  URL: https://www.w3.org/TR/rdf11-mt/


# ON REAL ALGEBRAIC LINKS IN THE $3$-SPHERE ASSOCIATED WITH MIXED POLYNOMIALS

RAIMUNDO N. ARAÚJO DOS SANTOS AND EDER L. SANCHEZ QUICENO

###### Abstract

Let $f:\mathbb{C}^{2}\to\mathbb{C}$ be a mixed polynomial, i.e., a complex
polynomial of variables $(u,\bar{u},v,\bar{v})$ such that $f(0)=0$ and $f$ has
an isolated singularity at the origin. Then, associated with $f$, we have the
well-defined link $L_{f}$, which is the intersection of the mixed hypersurface
$V_{f}:=f^{-1}(0)$ with a 3-sphere of small radius centered at the origin.
Such a link is called a real algebraic link. Classification and
characterization of real algebraic links are still open.

In this paper, we use properties of mixed polynomials to construct a new class
of real algebraic links. More exactly, real algebraic links arise from
products of mixed polynomials $f=p\cdot q:\mathbb{C}^{2}\to\mathbb{C}$, with
$p(0)=q(0)=0$.

Keywords: Isolated singularity; mixed polynomials; real algebraic links.

Acknowledgements: The authors are immensely thankful to Professor Osamu Saeki,
Director of Institute of Mathematics for Industry at Kyushu University, for
discussions and encouragements. This paper was developed during the visit of
second author at the Kyushu University, Japan, supported by grant
2019/11415-3, São Paulo Research Foundation (FAPESP).

##  1 INTRODUCTION

A link $L$ in $S^{3}$ is fibered if the complement $S^{3}\setminus L$ is a
surface bundle over the circle $S^{1}$ whose fiber is the interior of a
compact, oriented surface $F$ with $\partial F=L$. It is classically known
that fibered links in the sphere $S^{3}$ were characterized by Stallings in
[21]. It was shown that $L$ is fibered if and only if the commutator subgroup
of the fundamental group of the complement $\pi_{1}(S^{3}\setminus L)$ is
finitely generated. A very important class of links in $S^{3}$ are those that
arise associated with some type of singularities of polynomial germ maps
$(\mathbb{R}^{4},0)\to(\mathbb{R}^{2},0)$, in the following sense: a
polynomial map germ $f:(\mathbb{R}^{4},0)\to(\mathbb{R}^{2},0)$ has a _weakly
isolated singularity_ at the origin if we have $\Sigma(f)\cap V_{f}=\\{0\\}$
as a germ of set, where $\Sigma(f)$ is the singular set of $f$ and
$V_{f}:=f^{-1}(0)$. In the particular case where $\Sigma(f)=\\{0\\}$ we say
that $f$ has an _isolated singularity_ at the origin. A link $L$ in $S^{3}$ is
called _weakly algebraic_ (resp. _real algebraic_) if there exist a polynomial
map germ $f_{L}:(\mathbb{R}^{4},0)\to(\mathbb{R}^{2},0)$ with a weakly
isolated (resp. an isolated) singularity at the origin and a positive real
number $\rho_{0}$ such that $L_{f_{L},\rho}=V_{f_{L}}\cap S^{3}_{\rho}$ is
ambient isotopic to $L$ in $S^{3}_{\rho}\approx S^{3}$ for every
$0<\rho\leq\rho_{0}$, where $S^{3}_{\rho}$ is the sphere of radius $\rho$
centered at the origin in $\mathbb{R}^{4}$. In this case, such a polynomial
map germ $f_{L}$ is called a _weak realization_ (resp. _strong realization_)
of $L$.

Given a link $L$ in $S^{3}$ a weak realization is not a strong condition. In
fact, in [1] Akbulut and King have proven that every link $L$ in $S^{3}$
admits a weak realization $f_{L}:(\mathbb{R}^{4},0)\to(\mathbb{R}^{2},0)$.

Real algebraic links have a more restrictive property as pointed out by J.
Milnor in [16]. For a germ of complex isolated singularity
$g:(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ the classic result due to J. Milnor
[16] guarantees that there exists a small radius $\rho_{0}$ such that for all
$0<\rho\leq\rho_{0}$ the map

$\text{arg}(g)=\dfrac{g}{|g|}:S^{2n-1}_{\rho}\setminus L_{g,\rho}\to S^{1}$

is a smooth locally trivial fibration. The link $L_{g,\rho}=V_{g}\cap
S_{\rho}^{2n-1}$ is well-defined for all small radius $0<\rho\leq\rho_{0}$
because the manifolds $L_{g,\rho}$ are isotopic. Hence one denotes
$L_{g,\rho}$ by $L_{g}$.

In the case where $n=2$ and $g$ a germ of polynomial complex isolated
singularity, the link $L_{g}$ is called an _algebraic link_ and it was
completely characterized as union of iterated torus knots with some
restrictions in their characteristic pairs. For more details see [10, 11, 12,
13, 22].

A complex polynomial $f:(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ can be seen as a
real polynomial map $(Re(f),Im(f)):(\mathbb{R}^{2n},0)\to(\mathbb{R}^{2},0)$,
and for this reason, algebraic links are a well known class of real algebraic
links. For a real analytic map germ with an isolated singularity at the origin
$f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0),n>p\geq 2$, Milnor also proved that
there exists a small radius $\rho_{0}$ such that for all $0<\rho\leq\rho_{0}$
the link $L_{f,\rho}=V_{f}\cap S^{n-1}_{\rho}$ is always a fibered link.
Hence, for $n=4,p=2$ and $f$ a germ of polynomial map with an isolated
singularity the real algebraic link $L_{f}$ is a fibered link.

Classification and characterization for the real case are still open and in
Benedetti and Shiota [4] it was conjectured that: all fibered links are real
algebraic links.

A partial answer for this conjecture was provided by Looijenga in [15] where
as a consequence of a more general construction he proved that given a fibered
link $(S^{3},L)$ a special connected sum $(S^{3},L)\\#(S^{3},L)$ is an _odd
fibered link_. It means that, the fibered link and its fibration map are
invariant under the antipodal map
$\alpha(x_{1},x_{2},x_{3},x_{4})=(-x_{1},-x_{2},-x_{3},-x_{4}),$ and hence one
can find a strong realization of $(S^{3},L)\\#(S^{3},L).$

More recently, B. Bode [6] following a construction made in [9] and [18]
studied links $L$ in $S^{3}$ from a braid theoretic point of view, and proved
that given a _square braid_ $B$111$B=w^{2}$ for a braid $w$ on $s$ strands on
$s$ strands that closes $L$, one can find a mixed polynomial
$p:\mathbb{C}^{2}\to\mathbb{C}$ such that the real polynomial map
$p_{L}:=(Re(p),Im(p)):\mathbb{R}^{4}\to\mathbb{R}^{2}$ is a weak realization
of link $L$. Also, in the particular case where the braid $B$ is homogeneous,
then $p_{L}$ can be constructed as a strong realization of $L$. Thus, it is a
real algebraic link. For a brief summary of this construction see Section 2 of
this paper.

Other strong realizations of fibered links with special properties can be
found in the literature. For instance, L. Rudolph [20] constructed mixed
polynomials that are strong realizations of the figure-eight knot and
Borromean rings. A. Pichon [19] constructed strong realizations of the union
of iterated torus knots for mixed polynomials of type $f\cdot\bar{g},$ with
$f$ and $g$ complex polynomials without common branches. These classes of real
algebraic links are examples of fibered links that arise from strong
realizations of mixed polynomial and provides an interesting topic to be
studied.

In this paper we study mixed polynomials of type $f=p\cdot q$, where
$p:\mathbb{C}^{2}\to\mathbb{C}$ is a weak realization of the closure of a
square braid $B$ on $s$ strands and $q:\mathbb{C}^{2}\to\mathbb{C}$ is a non-
degenerate complex polynomial in the sense defined by Kouchnirenko in [14] or
in the general setting of mixed polynomials defined by M. Oka [17]. In Section
2, we summarize the construction of a special class of mixed polynomial and
prove a technical result that adds a necessary condition for our principal
result. In the Section 3 we study the Newton boundary of mixed polynomial
describing their faces relative to the Newton boundary. In Section 4 we prove
the principal result, Theorem 4.2, which says that a class of mixed
polynomials are strongly non-degenerate and convenient in the sense defined by
M. Oka [17]. We conclude with Proposition 4.4 that says that the links
associated with these mixed polynomials $f$ are real algebraic and their
Milnor fibrations are given by $\text{arg}(f):S^{3}_{\rho}\setminus L_{f}\to
S^{1}$. Also, we give an example of a strongly non-degenerate and convenient
mixed polynomial $f$ and we also use a characterization of non-degeneracy of
reducible complex polynomials due to E. Barroso et al. [2] to prove Corollary
4.7.

##  2 SEMIHOLOMORPHIC FUNCTIONS ASSOCIATED WITH BRAIDS

Let $L$ be a link in $S^{3}$ and $B$ a braid on $s$ strands that closes to
$L$. Section 2 contains a summary of how to construct a weak realization
$p:(\mathbb{C}^{2},0)\to(\mathbb{C},0)$ of $L$. The construction of this map
is due to B.Bode [6].

Consider the braid $B$ on $s$ strands as a union of $s$ disjoint parametric
curves $\gamma_{j}:[0,2\pi]\to\mathbb{R}^{2}\times[0,2\pi]$ given by

$\gamma_{j}(t)=(X_{j}(t),Y_{j}(t),t),$ (1)

where, each $X_{j}$ and each $Y_{j}$ is a smooth function
$[0,2\pi]\to\mathbb{R}$ and such that
$(X_{\pi_{B}(j)}(0),Y_{\pi_{B}(j)}(0))=(X_{j}(2\pi),Y_{j}(2\pi))$ for all
$j\in\\{1,2,...,s\\}$ and some permutation $\pi_{B}\in S_{s}$. For a
parameterization as in equation (1), we can define a family of polynomials in
complex variable $u$.

$\begin{array}[]{cc}&g_{a}:\mathbb{C}\times[0,2\pi]\to\mathbb{C}\\\ \\\
&g_{a}(u,t)=\prod_{j=1}^{s}(u-a(X_{j}(t)-iY_{j}(t))),\end{array}$ (2)

with $a>0$. By construction, the zero level set of $g_{a}$ is the braid $B$
for any choice of $a$. Let $\mathscr{C}$ be the set of link components of the
closure of $B$ and $s_{C}$ be the number of strands forming the link component
$C\in\mathscr{C}$.

B. Bode and M. R. Dennis [9] showed that if we consider the parameterization
(1) being a Fourier parameterization, i.e., for each $C\in\mathscr{C}$, the
$s_{C}$ curves that close $C$ are parameterizated by

$\gamma_{C,j}(t)=\left(F_{C}\left(\frac{t+2\pi
j}{s_{C}}\right),G_{C}\left(\frac{t+2\pi j}{s_{C}}\right),t\right),$ (3)

where $F_{C},G_{C}:\mathbb{R}\to\mathbb{R}$ are trigonometric polynomials

$F_{C}(t)=\sum^{N_{C}}_{l=-N_{C}}a_{C,l}e^{ilt},G_{C}(t)=\sum^{M_{C}}_{l=-M_{C}}b_{C,l}e^{ilt},$

with $a_{C,-l}=\overline{a_{C,l}}$ and $b_{C,-l}=\overline{b_{C,l}}$ for all
$C\in\mathscr{C}$ and all $l$. Then, the mapping $g_{a}$ can be written as a
polynomial in $u,e^{it}$ and $e^{-it}$.

${\mathbb{C}\times[0,2\pi]}$${\mathbb{C}\times
S^{1}}$${\mathbb{C}}$$\scriptstyle{g_{a}}$$\scriptstyle{Id\times
e^{it}}$$\scriptstyle{\Phi}$ (4)

In other words, there exists a map $\Phi$ that makes the diagram (4)
commutative; $g_{a}(u,t)=\Phi(u,e^{it})$. By abuse of notation, we continue to
write $g_{a}$ by $\Phi$. We summarize in this Section, how the map $g_{a}$ was
used for getting a polynomial map being a weak realization of $L$. The proof
of the following result can be found in [6, 9].

######  Theorem 2.1 (Lemma 4.1, B. Bode [6]).

Let $g_{a}$ constructed as in equation (2) from a Fourier parametrization (3)
of the square braid $B$ on $s$ strands. For large enough positive integer $k$
and small enough $a>0$

$p_{a,k}(u,v,\bar{v}):=(v\bar{v})^{sk}g_{a}(u/(v\bar{v})^{k},v/\sqrt{v\bar{v}}),$

satisfies:

  * 1.

The map $p_{a,k}$ is semiholomorphic polinomial in variable $u$, i.e., doesn’t
depend in $\bar{u}$, it’s monic in variable $u$ and $deg_{u}p_{a,k}=s$.

  * 2.

The map $p_{a,k}$ is a weak realization of the closure of $B$.

Hence, the closure of $B$ is weakly algebraic.

For a polynomial $p\in\mathbb{C}[u]$, we remind that the critical values
$v_{j}$ of $p$ are the images of critical roots, i.e., $v_{j}=p(u_{j})$, where
$u_{j}$ are the roots of $p_{u}\in\mathbb{C}[u]$ and $p_{u}(u)=\frac{\partial
p}{\partial u}(u)$. We define

$\displaystyle V_{s}=$ $\displaystyle\\{p\in\mathbb{C}[u]:p\text{ is monic, of
degree }s\text{ and has }s-1\text{ distinct critical values }$
$\displaystyle\text{ none of which are equal to zero or the constant term of
}p\\}.$

and

$W_{s}=\\{(v_{1},v_{2},...,v_{s-1})\in(\mathbb{C}\setminus\\{0\\})^{s-1}:v_{i}\neq
v_{j}\text{ for all }i\neq j\\}/S_{s-1},$

where $S_{s-1}$ is the symmetric group on $s-1$ elements. The following
theorem is due B. Bode [5] following previous results due to Beardon, Carne
and Ng [3].

######  Theorem 2.2 (Theorem 6.3, B. Bode [5]).

The map $\phi:V_{s}\to W_{s}$ that sends a polynomial $p\in V_{s}$ to its set
of critical values is a fibration.

B. Bode and M.R Dennis [9] proved that to find a Fourier parametrization of
any braid is always possible, using trigonometric interpolating data points
that can be obtained from a presentation of the braid diagram. We can see that
both the Fourier parametrization and the map $g_{a}$ can be constructed
satisfying a extra condition.

######  Lemma 2.3.

Let $B$ be a square of braid on $s$ strands. Then, there exist a Fourier
parametrization of $B$ and small $a>0$ such that satisfies:

  * 1.

The critical values $v_{j}(t),\ j=1,2,...,s-1$ of the polynomial in variable
$u$, $g_{a}$ are non-zero smooth function for all $j$, where

$\displaystyle g_{a}(u,e^{it})$
$\displaystyle=\prod_{C\in\mathscr{C}}\prod_{j=1}^{s_{C}}\left(u-a\left(F_{C}\left(\frac{t+2\pi
j}{s_{C}}\right)+iG_{C}\left(\frac{t+2\pi j}{s_{C}}\right)\right)\right),$

  * 2.

The image of $\psi_{0}:S^{1}\to\mathbb{C}$ defined by

$\psi_{0}(e^{it},e^{-it})=a^{s}\prod_{C\in\mathscr{C}}\prod_{j=1}^{s_{C}}\left(F_{C}\left(\frac{t+2\pi
j}{s_{C}}\right)+iG_{C}\left(\frac{t+2\pi j}{s_{C}}\right)\right)$

does not contain $0\in\mathbb{C}$.

  * 3.

For small enough $a>0$ and large integer $k>0$, the mixed polynomial $p_{a,k}$
is a weak realization of the closure of $B$.

###### Proof.

Let $B=\prod^{n}_{i=1}\sigma^{\epsilon_{i}}_{j_{i}}$ be the word braid of $B$.
We have two ways to obtain Fourier parametrization satisfying (2). First, we
can ask for that the trigonometric interpolating data points satisfying
$(F_{C}(t),G_{C}(t))\neq(0,0)$ for all components $C\in\mathscr{C}$ and
$t\in[0,2\pi]$. Second, let $(F_{C}(t),G_{C}(t))$ a Fourier parametrization of
the braid $B$, we have that there exists a non-zero complex number
$c=c_{1}+ic_{2}$ satisfying $(F_{C}(t)+c_{1},G_{C}(t)+c_{2})\neq(0,0)$ for all
components $C\in\mathscr{C}$ and $t\in[0,2\pi]$. It’s clear that
$(F_{C}(t)+c_{1},G_{C}(t)+c_{2})$ is a Fourier parametrization of the braid
$B$ and satisfy condition (2).

In other hand by Theorem 2.2 the critical values of the family of complex
polynomials

$g(u,e^{it})=\prod_{C\in\mathscr{C}}\prod_{j=1}^{s_{C}}\left(u-F_{C}\left(\frac{t+2\pi
j}{s_{C}}\right)+iG_{C}\left(\frac{t+2\pi j}{s_{C}}\right)\right)$

is given by smooth map $\phi\circ g(u,e^{it})$, that is clearly a smooth
functions since $F_{C},G_{C}$ and $\phi$ are smooth maps. The complex
polynomial $g_{a}(u,e^{it})$ has simply roots by the construction, so for all
roots $c_{j}(t)$ of partial derivative $g_{u}$ the critical value
$w_{j}(t)=g(c_{j}(t),e^{it}),\ j=1,...,s-1,$ are non-zero smooth complex
functions for all $j$. Define for $a>0$

$\displaystyle g_{a}(u,e^{it})$
$\displaystyle=\prod_{C\in\mathscr{C}}\prod_{j=1}^{s_{C}}\left(u-a\left(F_{C}\left(\frac{t+2\pi
j}{s_{C}}\right)+iG_{C}\left(\frac{t+2\pi j}{s_{C}}\right)\right)\right)$
$\displaystyle=a^{s}\prod_{C\in\mathscr{C}}\prod_{j=1}^{s_{C}}\left(u/a-\left(F_{C}\left(\frac{t+2\pi
j}{s_{C}}\right)+iG_{C}\left(\frac{t+2\pi j}{s_{C}}\right)\right)\right).$

Thus, $(g_{a})_{u}=a^{s-1}g_{u}(u/a,e^{it})$, note that the roots of
$(g_{a})_{u}$ are $ac_{j}(t),\ j=1,...,s-1$. Then, the non-zero critical
valued of $g_{a}$ are

$v_{j}(t)=g_{a}(ac_{j}(t),e^{it})=a^{s}g(c_{j}(t),e^{it})=a^{s}w_{j}(t),\
j=1,...,s-1.$

Therefore, we have that $v_{j}(t)$ are non-zero smooth complex functions for
all $j$ and by Theorem 2.1 we can choose $a>0$ small enough and a integer $k$
large enough such that
$p_{a,k}(u,v,\bar{v})=(v\bar{v})^{ks}g_{a}(u/(v\bar{v})^{k},v/\sqrt{v\bar{v}})$
satisfies (3). ∎

B. Bode in [5] characterized the singular points of the mixed polynomial
$p_{a,k}$ in terms of the map $g_{a}$ and in [6, 7, 8] described some families
of strong realizations of mixed polynomials.

######  Proposition 2.4.

The map $p_{a,k}$ has an isolated singularity if and only if $g_{a}$ has no
argument-critical points, which is equivalent to the non-vanishing of
$\frac{\partial arg(v_{j}(t))}{\partial t}$ for all $t\in[0,2\pi]$ and all
$j=1,2,...,n-1$, where $v_{j}(t),j=1,2,...,n-1,$ are the critical values of
the complex polynomial $g_{a}(u,e^{it})$.

To know when a mixed polynomial $\mathbb{C}^{2}\to\mathbb{C}$ has an isolated
singularity isn’t an easy task. But, there exist combinatorial techniques to
make that. We study in the next section a family of mixed polynomials
$f:\mathbb{C}^{2}\to\mathbb{C}$ with either weakly isolated singularity or an
isolated singularity.

##  3 NON-DEGENERATE AND STRONGLY NON-DEGENERATE MIXED POLYNOMIAL

Suppose that we are given a mixed polynomial
$f(z,\bar{z})=\sum_{j=1}^{l}c_{\nu_{j},\mu_{j}}z^{\nu_{j}}\bar{z}^{\mu_{j}},$
with $c_{0,0}=0$ so that $0\in f^{-1}(0)$, where $0$ is the origin of
$\mathbb{C}^{n}$. The zero set of $f$, $V_{f}=f^{-1}(0)$ is called the mixed
hypersurface. If $\mu_{j}=(0,...,0),\ j=1,\dots,l$, the mixed polynomial $f$
is simple a complex polynomial in variable $z$. Recall that a complex
polynomial $h(z)$ is called a _weighted homogeneous polynomial_ with weights
$P=\mathstrut^{t}(p_{1},...,p_{n})$ and type $(P;d)$, if $p_{1},...,p_{n}$ are
integers and there exists a positive integer $d$ so that

$f(t^{p_{1}}z_{1},...,t^{p_{n}}z_{n})=t^{d}f(z),t\in\mathbb{C}^{*}:$

The integer $d$ is called the degree of $f$ with respect to the weight vector
$P$. In the mixed case, a mixed polynomial $f$ is called a radially weighted
homogeneous polynomial if there exist integers $q_{1},...,q_{n}\geq 0$ and
$d_{r}>0$ such that it satisfies the equality:

$f(t^{q_{1}}z_{1},...,t^{q_{n}}z_{n},t^{q_{1}}\bar{z}_{1},...,t^{q_{n}}\bar{z}_{n})=t^{d_{r}}f(z,\bar{z}),\
t\in\mathbb{R}^{*}.$

Let $M=z^{\nu}\bar{z}^{\mu}$ be a mixed monomial where
$\nu=(\nu_{1},...,\nu_{n}),\ \mu=(\mu_{1},...,\mu_{n})$ and let
$P=\mathstrut^{t}(p_{1},\dots,p_{n})$ be a weight vector. We define the
_radial degree_ of $M$, $rdeg_{P}M$ with respect to $P$ by

$rdeg_{P}M=\sum^{n}_{j=1}p_{j}(\nu_{j}+\mu_{j}).$

Putting $Q=\mathstrut^{t}(q_{1},...,q_{n})$, $f$ be a radially weighted
homogeneous polynomial of type $(Q;d_{r})$ is equivalent to
$rdeg_{Q}z^{\nu_{j}}\bar{z}^{\mu_{j}}=d_{r}$ for $j=1,...,l$ with
$c_{\nu_{j},\mu_{j}}\neq 0$.

The radial Newton polygon $\Gamma_{+}(f;z,\bar{z})=\Gamma_{+}(S)$ (at the
origin) of a mixed polynomial $f$ is defined by the convex hull of

$\bigcup_{c_{\nu,\mu}\neq 0}(\nu+\mu)+(\mathbb{R}^{+})^{n},$

where $S=\\{\nu+\mu,c_{\nu,\mu}\neq 0\\}$. We call $\Gamma_{+}(f;z,\bar{z})$
or simply $\Gamma_{+}(f)$ the Newton polygon of $f$. The Newton boundary
$\Gamma(f;z,\bar{z})$ or simply $\Gamma(f)$ is defined by the union of compact
faces of $\Gamma_{+}(f)$. Observe that $\Gamma(f)$ is nothing but the ordinary
Newton boundary if $f$ is a complex polynomial. For a given positive integer
vector $P=\mathstrut^{t}(p_{1},\dots,p_{n})$, we associate a linear function
$l_{P}$ on $\Gamma(f)$ defined by
$l_{P}(\nu)=p_{1}\nu_{1}+\cdots+p_{n}\nu_{n}$ for
$\nu=(\nu_{1},\cdots,\nu_{n})\in\Gamma(f),$ and let $\Delta(P,f)=\Delta(P)$ be
the face where $l_{P}$ takes its minimal value. In other words, $P$ gives
radial weights for variables $z_{1},\dots,z_{n}$ by
$rdeg_{P}z_{j}=rdeg_{P}\bar{z}_{j}=p_{j}$ and
$rdeg_{P}z^{\nu}\bar{z}^{\mu}=\sum_{j=1}^{n}p_{j}(\nu_{j}+\mu_{j})$. We denote
the minimal value of $l_{P}$ by $d(P,f)$ or simply $d(P)$. Note that
$d(P,f)=min\\{rdeg_{P}z^{\nu}\bar{z}^{\mu}|c_{\nu,\mu}\neq 0\\}$, for a non-
negative weight $P$, we define the face function $f_{P}(z,\bar{z})$ by

$f_{\Delta(P)}(z,\bar{z})=f_{P}(z,\bar{z})=\sum_{\nu+\mu\in\Delta(P)}c_{\nu,\mu}z^{\nu}\bar{z}^{\mu}.$

Suppose that is given a mixed polynomial $f$. For non-negative weight vector
$P$, the face function $f_{P}(z,\bar{z})$ is a radially weighted homogeneous
polynomial of type $(p_{1},\dots,p_{n},d)$ with $d=d(P;f)$.

######  Definition 3.1.

Let $P$ be a positive weight vector. We say that $f$ is non-degenerate for
$P$, if the fiber $f_{P}^{-1}(0)\cap\mathbb{C}^{*n}$ contains no critical
point of the mapping $f_{P}:\mathbb{C}^{*n}\to\mathbb{C}$, where
$\mathbb{C}^{*n}=(\mathbb{C}\setminus\\{0\\})^{n}$. We say that $f$ is
strongly non-degenerate for $P$ if the mapping
$f_{P}:\mathbb{C}^{*n}\to\mathbb{C}$ has no critical points. A mixed
polynomial $f$ is called non-degenerate (resp. strongly non-degenerate) if $f$
is non-degenerate (resp. strongly non-degenerate) for any positive weight
vector $P$.

######  Remark 3.2.

Assume that $f$ is a holomorphic function. Then $f_{P}(z)$ is a weighted
homogeneous polynomial and we have the Euler equality:

$d(P,f)f_{P}(z)=\sum_{j=1}^{n}p_{j}z_{j}\frac{\partial f_{P}}{\partial
z_{j}}(z).$

Thus $f_{P}:\mathbb{C}^{*n}\to\mathbb{C}$ has no critical points over
$\mathbb{C}^{*}$. Thus $f$ is non-degenerate for $P$ implies $f$ is strongly
non-degenerate for $P$.

######  Proposition 3.3 (Proposition 6, M. Oka [17]).

Let $g(z,\bar{z})$ be a radially weighted homogeneous polynomial and let
$M:=z^{a}\bar{z}^{b}$ be a mixed monomial and put $h:=M\cdot g(z,\bar{z})$.
Then $0$ is a regular value of $g:\mathbb{C}^{*n}\to\mathbb{C}$ if and only if
$0$ is a regular value of $h:\mathbb{C}^{*n}\to\mathbb{C}$.

The assertion is immediate from the definition because
$g^{-1}(0)\cap\mathbb{C}^{*n}=h^{-1}(0)\cap\mathbb{C}^{*n}$ and the tangential
map $dh_{w}:T_{w}\mathbb{C}^{*n}\to T_{0}\mathbb{C}$ is equal to $Mdg_{w}$ for
any $w\in g^{-1}(0)\cap\mathbb{C}^{*n}$.

######  Lemma 3.4.

Let $f:\mathbb{C}^{n}\to\mathbb{C}$ be a non-degenerate mixed polynomial and
let $M=z^{a}\bar{z}^{b}$ be a mixed monomial. Then, the mixed polynomial
$h=M\cdot f$ is a non-degenerate mixed polynomial.

###### Proof.

Let $P$ be a positive weight vector, since $f$ is non-degenerate we have
$f_{P}:\mathbb{C}^{*n}\to\mathbb{C}$ has no critical points in
$f_{P}^{-1}(0)\cap\mathbb{C}^{*n}$, equivalently $0$ is regular value of
$f_{P}$. Then by Proposition 3.3 we have that $0$ is regular value of $M\cdot
f_{P}$ but we can easily to see that $h_{P}=M\cdot f_{P}$. Therefore, $h$ is a
non-degenerate mixed polynomial. ∎

######  Example 3.5.

Consider mixed polynomial $p_{a,k}$ that is a weak realization of the figure-
eight knot (the closure of $B=(\sigma_{1}\sigma_{2}^{-1})^{2}$) constructed by
B.Bode in [5].

$\displaystyle p_{a,k}$
$\displaystyle(u,v,\bar{v})=u^{3}+\frac{3}{4}a^{2}(v\cdot\bar{v})^{2k}u\left(\frac{v^{2}}{v\cdot\bar{v}}-\frac{\bar{v}^{2}}{v\cdot\bar{v}}\right)$
$\displaystyle-\frac{1}{8}a^{3}(v\cdot\bar{v})^{3k}\left(4\left(\frac{v^{2}}{v\cdot\bar{v}}+\frac{\bar{v}^{2}}{v\cdot\bar{v}}\right)+\left(\frac{v^{4}}{(v\cdot\bar{v})^{2}}-\frac{\bar{v}^{4}}{(v\cdot\bar{v})^{2}}\right)\right).$

$\boldsymbol{(1,4k)}$$\boldsymbol{(3,0)}$$\boldsymbol{(0,6k)}$$\boldsymbol{P=\mathstrut^{t}(4k,2)}$$\boldsymbol{Q_{1}=\mathstrut^{t}(4k,1)}$$\boldsymbol{Q_{2}=\mathstrut^{t}(4k,3)}$$\Gamma_{+}(p_{a,k})$
Figure 1: Newton polygon of weak realization of the figure-eight knot

The Newton boundary of $p_{a,k}$ is showed in figure 1, it consists of one
1-face joining the points $(3,0),(1,4k)$ and $(0,6k)$. The weight vector
associated with this face is $P=\mathstrut^{t}(4k,2)$ thus the face function
$(p_{a,k})_{P}(u,v,\bar{v})=p_{a,k}(u,v,\bar{v})$. We know by Theorem 2.1 that
$p_{a.k}$ has no critical points in $(p_{a,k})_{P}^{-1}(0)$, then $p_{a,k}$ is
non-degenerate for $P$.

We have only two other possibilities of face functions relative to positive
weight vectors. For instance, for the weight vectors
$Q_{1}=\mathstrut^{t}(4k,1)$ and $Q_{2}=\mathstrut^{t}(4k,3)$ we have the face
functions $(p_{a,k})_{Q_{1}}(u,v,\bar{v})=u^{3}$ and
$(p_{a,k})_{Q_{2}}(u,v,\bar{v})=-\frac{1}{8}a^{3}(v\cdot\bar{v})^{6k}\left(4\left(\frac{v^{2}}{v\cdot\bar{v}}+\frac{\bar{v}^{2}}{v\cdot\bar{v}}\right)+\left(\frac{v^{4}}{(v\cdot\bar{v})^{2}}-\frac{\bar{v}^{4}}{(v\cdot\bar{v})^{2}}\right)\right)$.

We can see that $(p_{a,k})_{Q_{1}}^{-1}(0)\cap\mathbb{C}^{*2}$ is empty, thus
$p_{a,k}$ is non-degenerate for $Q_{1}$. Take
$(p_{a,k})_{Q_{2}}(u,re^{it},re^{-it})=\frac{1}{8}r^{6k}\psi_{0}(t)$, where

$\displaystyle\psi_{0}(t)$
$\displaystyle=-4a^{3}\left(\frac{v^{2}}{v\cdot\bar{v}}+\frac{\bar{v}^{2}}{v\cdot\bar{v}}\right)-a^{3}\left(\frac{v^{4}}{(v\cdot\bar{v})^{2}}-\frac{\bar{v}^{4}}{(v\cdot\bar{v})^{2}}\right)$
$\displaystyle=-4a^{3}cos(2t)(2+isin(2t)).$ (5)

We have that the Jacobian matrix of $(p_{a,k})_{Q_{2}}$ for variables
$(u,|v|,arg(v))$ is given by

$\nabla(p_{a,k})_{Q_{2}}(u,v,\bar{v})=\left(\begin{array}[]{cccc}\begin{array}[]{l}0\end{array}&\begin{array}[]{l}0\end{array}&\begin{array}[]{l}\alpha\end{array}&\begin{array}[]{l}\beta\end{array}\\\
\begin{array}[]{l}0\end{array}&\begin{array}[]{l}0\end{array}&\begin{array}[]{l}\gamma\end{array}&\begin{array}[]{l}\delta\end{array}\par\end{array}\right).\\\
$

Where, $\alpha=\frac{\partial Re((p_{a,k})_{Q_{2}})}{\partial
r},\beta=\frac{\partial
Re((p_{a,k})_{Q_{2}})}{\partial\text{arg}(v)},\gamma=\frac{\partial
Im((p_{a,k})_{Q_{2}})}{\partial r}$ and
$\delta=\frac{\partial Im((p_{a,k})_{Q_{2}})}{\partial\text{arg }(v)}$. We
have that critical points of $(p_{a,k})_{Q_{2}}$ are the vanish set of
following determinant

$\displaystyle\alpha\delta-\beta\gamma$
$\displaystyle=\frac{3}{4}kr^{6k-1}Re(\psi_{0}(t))\delta-\frac{3}{4}kr^{6k-1}Im(\psi_{0}(t))\beta$
$\displaystyle=\frac{3}{32}kr^{12k-1}\left(Re(\psi_{0}(t))\frac{\partial
Im(\psi_{0}(t))}{\partial t}-Im(\psi_{0}(t))\frac{\partial
Re(\psi_{0}(t))}{\partial t}\right).$

By equation (3.5) we have

$Re(\psi_{0}(t))=-8a^{3}cos(2t)\text{ and }Im(\psi_{0}(t))=-2a^{3}isin(4t).$

Therefore, we can calculate the determinant

$\alpha\delta-\beta\gamma=\frac{3}{4}ka^{6}r^{12k-1}cos(2t)(3cos^{2}(2t)-1).$

Note that
$(u,re^{it})\in(p_{a,k})_{Q_{2}}^{-1}(0)\cap\Sigma((p_{a,k})_{Q_{2}})$, when
$cos(2t)=0$. Hence, $(p_{a,k})_{Q_{2}}$ has critical points in
$(p_{a,k})_{Q_{2}}^{-1}(0)\cap\mathbb{C}^{*2}$, thus $p_{a,k}$ is degenerate
for $Q_{2}$. Moreover, it’s no difficult to see that $p_{a,k}$ is a radially
weighted homogeneous polynomial of radial weight type $(6k,3;18k)$.

######  Remark 3.6.

We can see that all mixed polynomials $p_{a,k}$ defined after rescaling
process like Theorem 2.1 has a unique $1-$face, the line segment
$[(s,0),(0,2ks)]$. Moreover, the mixed polynomial,
$p_{a,k}(z,\bar{z})=\sum_{j=1}^{s}c_{\nu_{j},\mu_{j}}z^{\nu_{j}}\bar{z}^{\mu_{j}},$
where $z=(u,v)$, satisfies that: $\nu_{j}+\mu_{j}\in[(s,0),(0,2ks)]$ for
$c_{\nu_{j},\mu_{j}}\neq 0$.

The possibles pairs $\nu+\mu$ with $c_{\mu,\nu}\neq 0$ of $p_{a,k}$ are
$(s-j,2jk),\ j=0,\dots,s$. Hence, $p_{a,k}$ is a radially weighted homogeneous
polynomial of type $(2sk,s;2s^{2}k)$.

Consider a complex polynomial $q\in\mathbb{C}[u,v]$, with degree $m$ in $u$,
given by

$q(u,v)=u^{m}+\sum_{j=0}^{m-1}\theta_{j}(v)u^{j},\
\phi_{j}(v)\in\mathbb{C}[v]\text{ and }\text{ord}(\theta_{j})=n_{m-j}.$

######  Proposition 3.7.

Let $p_{a,k}:\mathbb{C}^{2}\to\mathbb{C}$ be a weak realization of the closure
of a square braid $B$ on $s$ strands, constructed as Lemma 2.3, and
$q:\mathbb{C}^{2}\to\mathbb{C}$ a complex polynomial. Then, there exists a
positive integer $k$, such that the Newton boundary of the mixed polynomial
$f:\mathbb{C}^{2}\to\mathbb{C}$, defined by
$f(u,v,\bar{v})=p_{a,k}(u,v,\bar{v})\cdot q(u,v)$, is formed by

$\Gamma(f)=\Gamma(f)^{1}\cup\Gamma(f)^{2},$

where,
$\Gamma(f)^{1}=\\{(\nu_{1},\nu_{2}+\mu_{2}+n_{m}):\nu+\mu\in\Gamma(p_{a,k})\\}$
and $\Gamma(f)^{2}=\\{(\nu_{1}+s,\nu_{2}):\nu\in\Gamma(q)\\}$.

###### Proof.

We know that $p_{a,k}$ is a radially weighted homogeneous polynomial of type
$(2sk,s;2s^{2}k)$ so the compact part of the Newton boundary of $p_{a,k}$ is
given by a unique line segment $[(s,0),(0,2ks)]$.

To study the Newton boundary of $f$ we only need to consider the points that
form the Newton boundary of $p_{a,k}$ and $q$, and to know how they contribute
to the Newton boundary of $f$.

Given a pair $(\nu_{1},\nu_{2})$ such that
$c_{\nu_{1},\nu_{2}}u^{\nu_{1}}v^{\nu_{2}}$ is a monomial of $q$. We take the
segments

$\ell_{1}=\Gamma(p_{a,k})+(\nu_{1},\nu_{2})=[(\nu_{1},2sk+\nu_{2}),(s+\nu_{1},\nu_{2})]$

and

$\ell_{2}=[(0,2sk+n_{m}),(s+\nu_{1},\nu_{2})]$

$\boldsymbol{\alpha}$$\boldsymbol{(0,2sk+n_{m})}$$\boldsymbol{\nu_{1}}$$\boldsymbol{(\nu_{1},2sk+\nu_{2})}$$\boldsymbol{s}$$\boldsymbol{(s,n_{m})}$$\boldsymbol{(s+\nu_{1},\nu_{2})}$$\boldsymbol{(s+m,0)}$Segment
$\Gamma(f)^{1}=\Gamma(p_{a,k})+(0,n_{m})$,Extension of
$\Gamma(p_{a,k})+(\nu_{1},\nu_{2})$ to axis $y$,
$c_{\nu_{1},\nu_{2}}u^{\nu_{1}}v^{\nu_{2}}$ appears in $q$,
Segment$[(s+\nu_{1},\nu_{2}),(0,2sk+n_{m})].$ Figure 2: Newton boundary of
$f=p_{a,k}\cdot q$

Let
$\alpha=(0,2k(s+\nu_{1})+\nu_{2}),\beta_{1}=(\nu_{1},\frac{2s^{2}k+n_{m}s+\nu_{1}\nu_{2}}{s+\nu_{1}}),\beta_{2}=(\nu_{1},2k(s-\nu_{1})+n_{m}),\theta_{1}=(s,2k\nu_{1}+\nu_{2})\text{
and }\theta_{2}=(s,\frac{2sk\nu_{1}+n_{m}\nu_{1}+s\nu_{2}}{s+\nu_{1}})$, where
$\alpha$ is the intersection between the extension of $\ell_{1}$ and the axis
$y$, and $\theta_{1}$ is the intersection of $\ell_{1}$ and the vertical line
$x=s$, $\beta_{1}\text{ and }\theta_{2}$ are the intersection of $\ell_{2}$
with the vertical line $x=\nu_{1}$ and $x=s$ respectively and $\beta_{2}$ is
the intersection of segment $[(0,2sk+n_{m}),(s,n_{m})]$ with the vertical line
$x=\nu_{1}$.

We have that $\Gamma(f)^{1}\subset\Gamma(f)$ if and only if the three
following conditions are hold for all $(\nu_{1},\nu_{2})$ with
$c_{\nu_{1},\nu_{2}}u^{\nu_{1}}v^{\nu_{2}}$ monomial of $q$:

  * 1.

$n_{m}<2k\nu_{1}+\nu_{2}$ and
$n_{m}<\frac{2sk\nu_{1}+n_{m}\nu_{1}+s\nu_{2}}{s+\nu_{1}}$,

  * 2.

$2k(s-\nu_{1})+n_{m}<2sk+\nu_{2}$ and
$2k(s-\nu_{1})+n_{m}<\frac{2s^{2}k+n_{m}s+\nu_{1}\nu_{2}}{s+\nu_{1}}$,

  * 3.

$2sk+n_{m}<2k(s+\nu_{1})+\nu_{2}$.

We can see that (1) and (2) are satisfied trivially for $k$ large as well the
first inequality in (2). It remains to prove that
$2k(s-\nu_{1})+n_{m}<\frac{2s^{2}k+n_{m}s+\nu_{1}\nu_{2}}{s+\nu_{1}}$ is
satisfied for $k$ large. A slight calculation show that the inequality is
equivalent to $-2k\nu_{1}^{2}+n_{m}\nu_{1}<\nu_{1}\nu_{2}$, that is trivially
satisfy for $k$ large. Hence, we choose $k$ such that (1), (2) and (3) hold
for all pairs $(\nu_{1},\nu_{2})$ with
$c_{\nu_{1},\nu_{2}}u^{\nu_{1}}v^{\nu_{2}}$ a monomial of $q$. Then,
$\Gamma(p_{a,k})+(0,n_{m})$ belongs to the Newton boundary of $f$ for $k$
large enough, i.e., $\Gamma(f)^{1}\subset\Gamma(f)$.

The remains of the Newton boundary of $f$ is formed by $1-$faces in
$\Gamma(f)$ joining the points $(s+m,0)$ and $(s,n_{m})$ these 1-faces are
simply the set of $1-$faces belonging to
$\Gamma(f)^{2}=\\{(\nu_{1}+s,\nu_{2}):\nu\in\Gamma(q)\\}$. Hence, the Newton
boundary of $f$ is the line segment $\Gamma(f)^{1}=[(s,n_{m}),(0,2sk+n_{m})]$
and $\Gamma(f)^{2}$. ∎

######  Corollary 3.8.

Let $f:\mathbb{C}^{2}\to\mathbb{C}$ as Proposition 3.7. Then, there exists a
positive integer $k$, such that the face functions relatives to all positive
weight vectors of the mixed polynomial
$f(u,v,\bar{v})=p_{a,k}(u,v,\bar{v})\cdot q(u,v)$ are:

$f_{P}=u^{s}q_{P}\text{ and }f_{Q}=v^{n_{m}}(p_{a,k})_{Q}$

for all $P,Q$ positive weight vectors.

###### Proof.

First, note that by Proposition 3.7 we can suppose unless to choose $k$
appropriately, that the Newton boundary of $f$ is given by
$\Gamma(f)=\Gamma(f)^{1}\cup\Gamma(f)^{2}$, where
$\Gamma(f)^{1}=\\{(\nu_{1},\nu_{2}+\mu_{2}+n_{m}):\nu+(0,\mu_{2})\in\Gamma(p_{a,k})\\}$
and $\Gamma(f)^{2}=\\{(\nu_{1}+s,\nu_{2}):\nu\in\Gamma(q)\\}.$

For a positive weight vector $P$ we have that the face $\Delta(P,p_{a,k})$ and
$\Delta(P,f)$ satisfy that

$f_{\Delta(P,f)}=f_{P}=v^{n_{m}}(p_{a,k})_{\Delta(P,p_{a,k})}=v^{n_{m}}(p_{a,k})_{P}.$

Analogously, for $\Delta(P,q)$ and $\Delta(P,f)$

$f_{\Delta(P,f)}=f_{P}=u^{s}q_{\Delta(P,q)}=u^{s}q_{P}.$

We have two classes of face functions of $f$. Namely, consider $P$ and $Q$ a
weight vectors such that $\Delta(P)\in\Gamma(q)$ and
$\Delta(Q)\in\Gamma(p_{a,k})$ respectively. Then $f_{P}=u^{s}q_{P}$ and
$f_{Q}=v^{n_{m}}(p_{a,k})_{Q}$. ∎

######  Example 3.9.

Let $q$ be a complex polynomial given by

$q(u,v)=u^{4}-2v^{3}u^{2}-4v^{5}u-v^{6}-v^{7}.$

The figure 3(b) shows the Newton boundary $\Gamma(q)$.

$\boldsymbol{(s+2,3)}$$\boldsymbol{(s+4,0)}$$\boldsymbol{(s,6)}$$\boldsymbol{(s,7)}$$\boldsymbol{(s+1,5)}$$\boldsymbol{(1,5+2sk)}$$\boldsymbol{(2,2sk+3)}$$\boldsymbol{(0,2sk+6)}$$\boldsymbol{(0,2sk+7)}$$\boldsymbol{(4,2sk)}$$\boldsymbol{P_{2}}$$\boldsymbol{P_{1}}$$\Gamma_{+}(f)$
(a) Newton boundary $\Gamma(f).$

$\boldsymbol{(2,3)}$$\boldsymbol{(4,0)}$$\boldsymbol{(0,6)}$$\boldsymbol{(0,7)}$$\boldsymbol{(1,5)}$$\boldsymbol{P=\mathstrut^{t}(3,2)}$$\Gamma_{+}(q)$
(b) Newton boundary $\Gamma(q)$

Figure 3: Newton boundaries of mixed polynomials $q$ and $f$.

Consider $f(u,v,\bar{v})=p_{a,k}(u,v,\bar{v})\cdot q(u,v)$ a mixed polynomial,
where $p_{a,k}$ is a weak realization of the square braid $B$ on $s$ strands.
Then, the Newton boundary is showed in figure 3(a).

The face functions of the mixed polynomial $f$ that aren’t simple monomials
are:
$f_{P_{1}}(u,v)=u^{s}(u^{4}-2v^{3}u^{2}-v^{6}),f_{P_{2}}(u,v,\bar{v})=v^{6}p_{a,k}(u,v,\bar{v})$
and $f_{Q}(u,v,\bar{v})=v^{6}\phi_{0}(v,\bar{v}),$ where $P_{1}$,$P_{2}$ and
$Q$ are positive weight vectors with $\Delta(Q)=(0,2sk+6)$ and $P_{1}$,
$P_{2}$ are showed in figure 3(a).

Note that, since $f_{P_{1}}$ is holomorphic complex polynomial we have that a
singular point satisfies
$(f_{P_{1}})_{u}(u,v)=u^{2}(7u^{4}-10v^{3}u^{2}-3v^{6})=0\text{ and
}(f_{P_{1}})_{v}(u,v)=-6u^{3}v^{2}(u^{2}+v^{3})=0.$ A simple calculation shows
that the conditions are valid only when $u=0$ or $v=0$. Therefore, $f_{P_{1}}$
has no critical points in $\mathbb{C}^{*2}$, thus $f$ is strongly non-
degenerate for $P_{1}$.

We will see in Theorem 4.2 that the strong non-degeneracy of $f$ for $P_{1}$
is also deduced for the non-degeneracy of the complex polynomial $q$. Hence,
the strong non-degeneracy of $f$ for $k$ large enough depend only in the class
of face functions $f_{P}=v^{6}(p_{a,k})_{P}$. Hence, Example 3.9 showed that
the class of mixed polynomials $f=p_{a,k}\cdot q$ with $q$ non-degenerate, is
a potential example of strongly non-degenerate mixed polynomial.

##  4 MILNOR FIBRATION AND STRONGLY NON-DEGENERATE AND CONVENIENT MIXED
POLYNOMIALS

Let
$f(z,\bar{z})=\sum_{j=1}^{l}c_{\nu_{j},\mu_{j}}z^{\nu_{j}}\bar{z}^{\mu_{j}}$ a
mixed polynomial. We say that $f$ is _convenient_ if for any $j=1,\dots,n$,
$\Gamma_{+}(f)$ intersects with $z_{j}$-axis. M.Oka [17] proved that if $f$ is
a non-degenerate and convenient mixed polynomial, there exists a positive
number $\rho_{0}$ such that $V_{f}$ is mixed non-singular except at the origin
in the ball $B^{2n}_{\rho_{0}}$ and the sphere $S^{2n-1}_{\rho}$ intersects
transversely with $V_{f}$ for every radius $\rho\leq\rho_{0}$. Moreover, when
we consider a strongly non-degenerate mixed polynomial has been proven.

######  Theorem 4.1 (Lemma 28, Oka [17]).

Assume that $f$ is a strongly non-degenerate mixed polynomial. Then, there
exist a positive numbers $\rho_{0}$ and $\delta$ such that the fiber
$V_{f,\eta}:=f^{-1}(\eta)$ has no mixed singularity in the ball
$B^{2n}_{\rho_{0}}$ for any non-zero $\eta$ with $|\eta|\leq\delta$.

Let $f$ be a strongly non-degenerate and convenient mixed polynomial. Then, by
Theorem 4.1 the isolatedness of the singularity at the origin of $f$ follows
from the convenience. Thus, mixed polynomials with these properties are strong
realizations of their associated links.

######  Theorem 4.2.

Let $B$ be a square braid on $s$ strands and
$p_{a,k}:\mathbb{C}^{2}\to\mathbb{C}$ be a weak realization of the closure of
$B$, constructed as Lemma 2.3. Then, there exist positive integers
$n^{\prime},k$ such that for all non-degenerate complex polynomial
$q:\mathbb{C}^{2}\to\mathbb{C}$ given by

$q(u,v)=u^{m}+\sum_{j=0}^{m-1}\theta_{j}(v)u^{j},\theta_{j}(v)\in\mathbb{C}[v]\text{
and }\text{ord}(\theta_{j})=n_{m-j}.$

with $n_{m}\geq n^{\prime}$, the mixed polynomial
$f:\mathbb{C}^{2}\to\mathbb{C}$, defined by
$f(u,v,\bar{v})=p_{a,k}(u,v,\bar{v})\cdot q(u,v)$ is strongly non-degenerate
and convenient mixed polynomial.

###### Proof.

Take a large integer $k$ satisfying Corollary 3.8. Then the face functions of
$f$ relatives to positive weight vectors are $f_{P}=u^{s}q_{P}\text{ and
}f_{Q}=v^{n_{m}}(p_{a,k})_{Q}$ for $P$ and $Q$ positive weight vectors. We are
going to prove the strong non-degeneracy of $f$ in two parts:

For the first class of face functions, by remark 3.2 since $q$ is a non-
degenerate holomorphic function then $q_{P}$ has no critical points in
$\mathbb{C}^{*2}$. Thus, by Lemma 3.4 and again remark 3.2 $f_{P}$ has no
critical points in $\mathbb{C}^{*2}$ as well. Therefore, $f$ is strongly non-
degenerate for $P$.

For the second class of face functions, since $p_{a,k}$ is a semiholomophic
and monic polynomial in variable $u$, we can write it as

$p_{a,k}(u,v,\bar{v})=u^{s}+\sum_{j=0}^{s-1}\phi_{j}(v,\bar{v})u^{j},\
\phi_{j}\in\mathbb{C}[v,\bar{v}].$

Moreover, $p_{a,k}$ is radially weighted homogeneous thus it remains to
consider just two face functions, associated with the weight vectors
$Q_{1}=(3,2sk)$ and $Q_{2}$ with $\Delta(Q_{2})=(2sk+n_{m},0)$, these are

$f_{Q_{1}}(u,v,\bar{v})=v^{n_{m}}p_{a,k}(u,v,\bar{v})\text{ and
}f_{Q_{2}}(u,v,\bar{v})=v^{n_{m}}\phi_{0}(v,\bar{v}).$

We know from the construction of $p_{a,k}$ that
$\phi_{0}(re^{it},re^{-it})=r^{2ks}\psi_{0}(e^{it},e^{-it})$, where
$\psi_{0}:S^{1}\to\mathbb{C}$ is a smooth map,
$\psi_{0}\in\mathbb{C}[e^{it},e^{-it}]$ and by (2) in Lemma 2.3 we can suppose
that the image of $\psi_{0}$ does not contain zero and we can consider
$\text{arg}(\psi_{0})$. Thus, we can study the map $\text{arg}(f_{Q_{2}})$
together with $\text{arg}(\psi_{0})$, in order to know the critical points of
$f_{Q_{2}}$.

$\displaystyle\text{arg}(f_{Q_{2}})(re^{it},re^{-it})$
$\displaystyle=-Re\big{(}ilog((re^{it})^{n_{m}}r^{2ks}\psi_{0}(e^{it},e^{-it})\big{)}$
$\displaystyle=n_{m}t+\text{arg}(\psi_{0})(e^{it},e^{-it}).$

Then, analogously to the Proposition 2.4, we can prove that the map
$f_{Q_{2}}$ has no critical points in $\mathbb{C}^{*2}$ if and only if
$\text{\text{arg}}(f_{Q_{2}})$ has no critical points in $\mathbb{C}^{*2}$.
Thus, we have to show that

$\frac{\partial\text{arg}(f_{Q_{2}})}{\partial t}(re^{it},re^{-it})\neq
0\text{ for }t\in[0,2\pi]\text{ and }r>0.$

Calculating the derivative, we have

$\displaystyle\frac{\partial\text{arg}(f_{Q_{2}})}{\partial
t}(re^{it},re^{-it})=n_{m}+\frac{\partial\text{arg}(\psi_{0})}{\partial
t}(e^{it},e^{-it}).$

We know by the continuity of the map
$\frac{\partial\text{arg}(\psi_{0})}{\partial t}:S^{1}\to\mathbb{R}$ that its
image is a bounded set of $\mathbb{R}$, thus there exists an integer
$\tilde{n}_{0}$ such that

$\tilde{n}_{0}+\frac{\partial\text{arg}(\psi_{0})}{\partial
t}(e^{it},e^{-it})\neq 0\text{ for }t\in[0,2\pi].$

Therefore, for $n_{m}\geq\tilde{n}_{0}$, $f$ is strongly non-degenerate for
$P$. Finally, consider $f_{Q_{1}}(u,v,\bar{v})=v^{n_{m}}p_{a,k}(u,v,\bar{v})$.
Let $v_{j}(t),\ j=1,2,...,s-1,$ be the non-zero critical values of the complex
polynomial $f_{Q_{1}}$, again by Proposition 2.4 we have that $f_{Q_{1}}$ has
no critical points in $\mathbb{C}^{*2}$ if and only if $f_{Q_{1}}$ has no
argument-critical points in $\mathbb{C}^{*2}$.

$\text{arg}(f_{Q_{1}})(u,re^{it})=n_{m}t+\text{arg}(p_{a,k})(u,re^{it})=n_{m}t+\text{arg}(g_{a})(u/r^{2k},e^{it}).$

Thus, the derivative

$\frac{\partial\text{arg}(f_{Q_{1}})}{\partial
t}(u,re^{it})=n_{m}+\frac{\partial\text{arg}(g_{a})}{\partial
t}(u/r^{2k},e^{it}).$

We know that $v_{j}(t)$ is non-zero smooth function for all $j=1,\dots s-1$ so
the image of $\frac{\partial\text{arg}(v_{j}(t))}{\partial t}$ is a bounded
set of $\mathbb{R}$, thus for each $j=1,2,...,s-1$ there exists an integer
$\tilde{n}_{j}$ such that
$\tilde{n}_{j}+\frac{\partial\text{arg}(v_{j}(t))}{\partial t}\neq 0$ for all
$t\in[0,2\pi]$. Hence, taking $n^{\prime}=\max\\{\tilde{n}_{0},\tilde{n}_{j}|\
j=1,\dots,s-1\\}$ we have that for $n_{m}\geq n^{\prime}$ the map $f_{Q_{1}}$
has no critical points in $\mathbb{C}^{*2}$. Hence, $f$ is a strongly non-
degenerate and convenient mixed polynomial for all $n_{m}\geq n^{\prime}$. ∎

We consider now the original Milnor fibration on the sphere, which is defined
as follows: $\text{arg}(f):S^{2n-1}_{\rho}\setminus K_{\rho}\to
S^{1},z\to\text{arg}(f)(z)=f(z,\bar{z})/|f(z,\bar{z})|,$ where
$K_{\rho}=V_{f}\cap S_{\rho}^{2n-1}$. First, we identify $\text{arg}(f)$ with
$\text{arg}(f)(z)=-Re(i\text{log}f(z))$, taking the argument $\theta$ as a
local coordinate of the circle $S^{1}$. M. Oka showed that strongly non-
degenerate and convenient mixed polynomials satisfy:

######  Theorem 4.3 (Theorem 33, M.Oka[17]).

Let $f:(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ be a strongly non-degenerate and
convenient mixed polynomial. Then, there exists a positive number $\rho_{0}$
such that

$\text{arg}(f):S^{2n-1}_{\rho}\setminus L_{f,\rho}\to S^{1}$

is a locally trivial fibration for every radius $\rho\leq\rho_{0}$. Moreover,
$f$ is a strong realization of $L_{f}$.

######  Corollary 4.4.

Let $p_{a,k}$ be a mixed polynomial as Theorem 4.2. Then, there exist positive
integers $n^{\prime},k$ and a positive number $\rho_{0}$ such that for every
non-degenerate complex polynomial $q:\mathbb{C}^{2}\to\mathbb{C}$ given by

$q(u,v)=u^{m}+\sum_{j=0}^{m-1}\theta_{j}(v)u^{j},\theta_{j}(v)\in\mathbb{C}[v]\text{
and }\text{ord}(\theta_{j})=n_{m-j}.$

with $n_{m}\geq n^{\prime}$, the link $L_{f}$ associated with $f=p_{a,k}\cdot
q$ is real algebraic. Moreover, the mapping
$\text{arg}(f):S^{3}_{\rho}\setminus L_{f,\rho}\to S^{1}$ is a locally trivial
fibration for every radius $\rho\leq\rho_{0}$.

###### Proof.

By Theorem 4.2 there exist positive integers $k,n^{\prime}$ such that for all
positive integers $n_{m}$ with $n_{m}\geq n^{\prime}$, the mixed polynomial
$f$ is a strongly non-degenerate mixed polynomial. It’s not difficult to see
that $f$ is a convenient mixed polynomial. Hence, by Theorem 4.1 we have that
$f$ has an isolated singularity at the origin. Moreover, the link $L_{f}$
associated with this isolated singularity of $f$ is a real algebraic link. By
Theorem 4.3, the strong non-degeneracy of $f$ implies the original Milnor
fibration of $f$, i.e., there exist a positive number $\rho_{0}$ such that
$\text{arg}(f):S^{3}_{\rho}\setminus L_{f,\rho}\to S^{1}$ is a locally trivial
fibration for every radius $\rho\leq\rho_{0}$. ∎

######  Example 4.5.

Consider $q:\mathbb{C}^{2}\to\mathbb{C}$ given by $q(u,v)=u^{2}+v^{n}$ and
$p_{a,k}$ a weak realization of the closure of
$B=(\sigma_{1}^{-1}\sigma_{2}\sigma_{1}^{3}\sigma_{2})^{2}$ (square of knot
$5_{2}$, figure 4(b)), constructed as Lemma 2.3. Consider a Fourier
parametrization of the braid $B$, figure 4(a), given for the trigonometric
functions

$(F_{C}(t),G_{C}(t))=\left(cos(4t/3)+\frac{3}{4}sin(10t/3),-sin(8t/3)-\frac{1}{2}cos(2t/3)\right).$

(a) Braid $(\sigma_{1}^{-1}\sigma_{2}\sigma_{1}^{3}\sigma_{2})^{2}$.

(b) Square of $5_{2}$ knot.

Figure 4: Square braid from a Fourier parametrization of the trigonometric
functions $(F_{C}(t),G_{C}(t))$.

Using equation (2) we calculate the complex polynomial $g_{a}(u,e^{it})$,

$\displaystyle
g_{a}(u,e^{it})=u^{3}+a^{2}u\bigg{(}-\frac{15}{64}+\frac{9}{16}ie^{-2it}-\frac{21}{16}ie^{2it}+\frac{15}{32}e^{-4it}-\frac{15}{32}e^{4it}$
$\displaystyle-\frac{9}{16}ie^{-6it}-\frac{9}{16}ie^{6it}\bigg{)}+a^{3}\bigg{(}\frac{53}{128}ie^{-2it}+\frac{47}{128}ie^{2it}$
$\displaystyle-\frac{85}{128}e^{-4it}-\frac{43}{128}e^{4it}+\frac{51}{256}ie^{-6it}-\frac{3}{8}-\frac{93}{256}ie^{6it}$
$\displaystyle+\frac{43}{128}e^{-8it}+\frac{11}{128}e^{8it}-\frac{27}{512}ie^{-10it}+\frac{27}{512}ie^{10it}$
$\displaystyle\bigg{)}.$

Then, the mixed polynomial $p_{a,k}$ is given by

$\displaystyle
p_{a,k}(u,v,\bar{v})=(v\bar{v})^{3k}g_{a}(u/(v\bar{v})^{k},v/\sqrt{v\bar{v}}).$

Consider the mixed polynomial
$f(u,v,\bar{v})=p_{a,k}(u,v,\bar{v})\cdot(u^{2}+v^{n})$, so the Newton polygon
$\Gamma_{+}(f)$ is showed in figure 5.

$\boldsymbol{(5,0)}$$\boldsymbol{(3,n)}$$\boldsymbol{(0,6k+n)}$$\boldsymbol{P_{2}}$$\boldsymbol{P_{1}}$$\Gamma_{+}(f)$
Figure 5: $\Gamma_{+}(f)$.

(a) $\frac{\partial\text{arg}(\phi_{a,0})}{\partial t}(re^{it},re^{-it}).$

(b) $\frac{\partial\text{arg}(v_{j}(t))}{\partial t},\ j=1,2.$

Figure 6: Derivative of argument maps for the face functions $f_{Q}$ and
$f_{P_{1}}$.

The face functions of $f$ for $P_{1}$,$P_{2}$ and $Q$ are:
$f_{P_{1}}(u,v)=u^{3}(u^{2}+v^{n})$,
$f_{P_{2}}(u,v,\bar{v})=v^{n}p_{a,k}(u,v,\bar{v})$ and
$f_{Q}(u,v,\bar{v})=v^{n}\phi_{a,0}(v,\bar{v}),$ where $\Delta(Q)=(0,6k+n)$.
We can prove directly that $f_{P_{1}}$ has no critical points in
$\mathbb{C}^{*2}.$ We can see in figure 6 that, for $n\geq 14$, we have
$f_{Q}$ and $f_{P_{2}}$ has no argument critical points in $\mathbb{C}^{*2}$.
Hence, for $n\geq 14$, $f$ is a strongly non-degenerate and convenient mixed
polynomial.

Take $n=14$, then unless to choice small enough $a>0$ and large $k$,
$f(u,v,\bar{v})=p_{a,k}(u,v,\bar{v})\cdot(u^{2}+v^{14})$ is a strong
realization of the closure of the braid on $5$ strands in figure 7.

Figure 7: Braid representation of $L_{f}$ with
$(u,t)\in\mathbb{C}\times[0,2\pi]$ when $v=r_{0}e^{it}$.

As a last discussion, we can present a criteria due to E. Barroso et al. [2]
to know when a complex polynomial is non-degenerate with respect to their
Newton boundary, thus we have a way to construct real algebraic links.

Given a reducible complex polynomial $q\in\mathbb{C}[u,v]$, without common
factors. The complete characterization of non-degeneracy of a polynomial
complex isolated singularity $q$, it is in terms of intrinsic topological
invariants of $q$. Let $\Gamma_{+}(q)$ the Newton polygon of $q$, then we can
associated a number $\nu(\Gamma_{+}(q))\in\mathbb{N}\cup{\infty}$ or simple
$\nu(q)$ called a Newton number of $q$, defined by Kouchnirenko in [14].

Consider that $q$ doesn’t have multiple factor, then the invariant $\mu(q)$ is
called the Milnor number of the reduced polynomial $q$. By definition,

$\mu(f)=\text{dim }\dfrac{\mathbb{C}[u,v]}{\left(\frac{\partial q}{\partial
u},\frac{\partial q}{\partial v}\right)}.$

The result in [2] that is a refined version of Kouchnirenko’s theorem in two
dimensions.

######  Theorem 4.6 (Theorem 4.1, E. Barroso et al.[2]).

Let $q:\mathbb{C}^{2}\to\mathbb{C}$ be a reduced polynomial without common
factors. Then $\mu(q)\geq\nu(q)$ with equality holding if and only if $q$ is
non-degenerate.

Now, we can enunciate Corollary 4.4 in terms of Newton number and Milnor
number of a reducible complex polynomial.

######  Corollary 4.7.

Let $p_{a,k}$ be a weak realization of the closure of a square braid $B$ on
$s$ strands, constructed as Lemma 2.3. Then, there exist positive integers
$n^{\prime},k$ and a positive number $\rho_{0}$ such that for every non-
degenerate complex polynomial $q:\mathbb{C}^{2}\to\mathbb{C}$ given by

$q(u,v)=u^{m}+\sum_{j=0}^{m-1}\theta_{j}(v)u^{j},\theta_{j}(v)\in\mathbb{C}[v]\text{
and }\text{ord}(\theta_{j})=n_{m-j},$

with $\nu(q)=\mu(q)$ and $n_{m}\geq n^{\prime}$, the link $L_{f}$ associated
with $f=p_{a,k}\cdot q$ is real algebraic. Moreover, the mapping
$\text{arg}(f):S^{3}_{\rho}\setminus L_{f,\rho}\to S^{1}$ is a locally trivial
fibration for every radius $\rho\leq\rho_{0}$.

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Eder Leandro Sanchez Quiceno (ICMC-USP/São Carlos)
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Raimundo Nonato Araújo dos Santos (ICMC-USP/São Carlos)
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