6 research outputs found
On Ritt's decomposition Theorem in the case of finite fields
A classical theorem by Ritt states that all the complete decomposition chains
of a univariate polynomial satisfying a certain tameness condition have the
same length. In this paper we present our conclusions about the generalization
of these theorem in the case of finite coefficient fields when the tameness
condition is dropped.Comment: 13 pages. v2: added comment from reader and additional reference
Résolution de certaines équations diophantiennes et propriétés de certains polynômes
Dans les quatre premiers chapitres de cette thèse, nous abordons quelques équations diophantiennes et leurs solutions. On démontre que l'équation y 2 = px(Ax2 + 2) n'admet qu'un maximum de six solutions entières où p est nombre premier et A > 1 est entier impair ; on démontre que l'équation Resx P(x), x2 + sx + t = a n'admet qu'un nombre ni de solutions (s, t) pour P un polynôme xe et a un entier autre que zéro ; on résout l'équation Fn−Fm = y a lorsque y ∈ {6, 11, 12} et on trouve une borne pour les solutions de Fn + Fm = y a dans le cas général ; et on démontre que si un nombre su sant d'entiers x consécutifs existent tels que P(x) est sous la forme mq lorsque q ≥ 2 est diviseur de deg P, alors P = Rq pour un certain polynôme R, ce qui nous permet de déduire l'existence d'une in nité de solutions à y q = P(x) à partir d'un nombre ni de telles solutions dans certains cas. Dans les six derniers chapitres, nous abordons plusieurs sujets reliés à la décomposition d'objets algébriques. Parmi les résultats, on présente quelques conditions sous lesquelles un polynôme ne peut pas être exprimé comme une composition de deux polynômes de degré inférieur ; on présente une nouvelle démonstration du théorème Carltiz-Lutz sur les polynômes de permutations ; on étudie la possibilité d'exprimer un polynôme comme une somme composée ou un produit composé de deux autres polynômes de degré inférieur ; on trouve une borne pour un des plus petits nombres premiers qui se décompose dans un corps imaginaire quadratique donné ; et on étudie la possibilité de recouvrir un anneau avec ses sous-anneauxThe rst four chapters of this thesis address some Diophantine equations and their solutions. We prove that the equation y 2 = px(Ax2 + 2) has at most six integer solutions (x, y) for p a prime and A > 1 an odd integer; we prove that the equation Resx P(x), x2 + sx + t = a has only nitely many integer solutions (s, t) for a xed polynomial P and nonzero integer a; we completely solve the equation Fn − Fm = y a for y ∈ {6, 11, 12} and bound the solutions for Fn + Fm = y a in general; and we prove that the existence of su ciently many consecutive integers x such that P(x) is of the form mq for q ≥ 2 dividing deg P implies that Rq for some polynomial R, providing criteria for deducing the existence of in nitely many solutions to y q = P(x) from the existence of nitely many solutions in some cases. In the last six chapters, we address various algebraic decomposition related topics. Among other results, we provide criteria which guarantee a polynomial cannot be written as a composition of two polynomials of smaller degree; we provide a new proof of the Carlitz-Lutz theorem on permutation polynomials; we study the possibility of expressing a polynomial as the composed sum or composed multiplication of two polynomials of smaller degree; we bound from below some of the smallest primes which split in an imaginary quadratic eld; and we study the possibility of covering a ring with its subring
Poles of the resolvent
M.Sc. (Mathematics)Any sensible piece of writing has an intended readership. Conversely, any piece of writing that has no intended readership has no sense. These are axioms of authorship and necessary directions to any prospective author. The aim of this dissertation was to serve as an experimental exposition of the analysis of the resolvent operator. Its intended readership is therefore graduate-level students in operator theory and Banach algebras. The analysis included in this dissertation is of a specific kind: it includes and occasionally extends beyond the analysis of a function at certain of its singularities of finite order. The exposition is experimental in the sense that it does not even aim at a comprehensive review of analysis of the resolvent operator, but it is concerned with that part of it which seems to have interesting and useful results and which appears to be the most suggestive of further research. In order to obtain an exhaustive exposition, we still lack a study of the properties of the resolvent operator where it is differentiable (which seemingly entails little more than undergraduate-level complex analysis), and a study of essential singularities of the resolvent operator (which seems too difficult for the expository style). A brief overview of the contents of this dissertation is in order: a chapter introducing some analytic concepts used throughout this dissertation; a chapter on poles of order 1 follows (so-called simple poles), where the Gelfand theorem (2.1.1) is the most important result; a chapter on poles of higher order, where the Hille theorem is the most prominent; and lastly some topics that have arisen out of the study of poles of the resolvent, collected in chapter 4. I should make it abundantly clear to the reader that although this dissertation is my work, it does not for the most part follow that the result are my own. What is my own is the arrangement, but as it is a literature study, the results are mainly those of other authors. My own addition has been mostly notes, usually in italics. The literature study has benefited very much from Zemanek's paper (Zemanek,[54]), and I am deeply indebted to him for it. Incidentally, this has also been a chance to exhibit my style of citation; the number corresponds to the number of the citation in the bibliography. There are numerous instances where I have indicated possible extensions and recumbent studies that could be roused effectively, but which have swelled this volume unnecessarily. For instance, the last subsection is little more than such indications
Integer values of polynomials
Let be a polynomial with rational coefficients, be an
infinite subset of the rational numbers and consider the image set
. If is a polynomial such that we say that
\emph{parametrizes} the set . Besides the obvious solution
we may want to impose some conditions on the polynomial ;
for example, if we wonder if there exists a
polynomial with integer coefficients which parametrizes the set
.
Moreover, if the image set is parametrized by a polynomial
, there comes the question whether there are any relations
between the two polynomials and . For example, if is a
linear polynomial and if we set , the polynomial
obviously parametrizes the set f(\Q). Conversely, if we have
f(\Q)=g(\Q) (or even ) then by Hilbert's
irreducibility theorem there exists a linear polynomial such
that . Therefore, given a polynomial which
parametrizes a set , for an infinite subset of the
rational numbers, we wonder if there exists a polynomial such
that . Some theorems by Kubota give a positive answer
under certain conditions.
The aim of this thesis is the study of some aspects of these two
problems related to the parametrization of image sets of
polynomials.
\bigskip
In the context of the first problem of parametrization we consider
the following situation: let be a polynomial with rational
coefficients such that it assumes integer values over the integers.
Does there exist a polynomial with integer coefficients such
that it has the same integer values of over the integers?
This kind of polynomials are called \emph{integer-valued}
polynomials. We remark that the set of integer-valued polynomials
strictly contains polynomials with integer coefficients: take for
example the polynomial , which is integer-valued over the
set of integers but it has no integer coefficients. So, if is an
integer-valued polynomial, we investigate whether the set
can be parametrized by a polynomial with integer coefficients; more
in general we look for a polynomial , for
some natural number , such that . In this
case we say that is -\emph{parametrizable}.
In a paper of Frisch and Vaserstein it is proved that the subset of
pythagorean triples of is parametrizable by a single triple
of integer-valued polynomials in four variables but it cannot be
parametrized by a single triple of integer coefficient polynomials
in any number of variables. In our work we show that there are
examples of subset of parametrized by an integer-valued
polynomial in one variable which cannot be parametrized by an
integer coefficient polynomial in any number of variables.
If is an integer-valued polynomial, we give the following
characterization of the parametrization of the set : without
loss of generality we may suppose that has the form ,
where is a polynomial with integer coefficients and is a
minimal positive integer. If there exists a prime different from
such that divides then is not
-parametrizable. If and is -parametrizable
then there exists a rational number which is the ratio of
two odd integers such that . Moreover
for some if and only if or
there exists an odd integer such that . We
show that there exists integer-valued polynomials such that
is -parametrizable with a polynomial
, but for every
.
\bigskip
In 1963 Schinzel gave the following conjecture: let be an
irreducible polynomial with rational coefficients and let be an
infinite subset of \Q with the property that for each in
there exists in such that ; then either is
linear in or is symmetric in the variables and .
We remark that if a curve is defined by a polynomial with Schinzel's
property then its genus is zero or one, since it contains infinite
rational points; here we use a theorem of Faltings which solved the
Mordell conjecture (if a curve has genus greater or equal to two
then the set of its rational points is finite). We will focus our
attention on the case of rational curves (genus zero). Our objective
is to describe polynomials with Schinzel's property whose
curve is rational and we give a conjecture which says that these
rational curves have a parametrization of the form
.
This problem is related to the main topic of parametrization of
image sets of polynomials in the following way: if
is a parametrization of a curve
(which means ), where is a polynomial
with Schinzel's property, let be the set
of the definition of Schinzel, where S'\subset\Q. Then for each
there exists such that ,
hence . So, in the case of rational
curves, the problem of Schinzel is related to the problem of
parametrization of rational values of rational functions with other
rational functions (we will show that under an additional hypothesis
we can assume that are polynomials). In
particular, if is a parametrization of a
curve defined by a symmetric polynomial, then
, where is an involution (that is
). So in the case of rational symmetric plane curves we
have this classification in terms of the parametrization of the
curve.
It turns out that this argument is also related to Ritt's theory of
decomposition of polynomials. His work is a sort of "factorization"
of polynomials in terms of indecomposable polynomials, that is
non-linear polynomials such that there are no of degree
less than such that . The indecomposable
polynomials are some sort of "irreducible" elements of this kind of
factorization.
\vspace{1.5cm}
% 1o capitolo
In the first chapter we recall some basic facts about algebraic
function fields in one variable, the algebraic counterpart of
algebraic curves. In particular we state the famous Luroth's
theorem, which says that a non trivial subextension of a purely
trascendental field of degree one is purely trascendental.
We give the definition of minimal couple of rational functions that
we will use later to characterize algebraically a proper
parametrization of a rational curve. We conclude the chapter with
the general notion of valuation ring of a field and we characterize
valuation rings of a purely trascendental field in one variable
(which corresponds geometrically to the Riemann sphere, if for
example the base field is the field of the complex numbers).
Moreover valuation rings of algebraic function fields in one
variable are discrete valuation rings.
\bigskip
%2o capitolo
In the second chapter we state the first theorem of Ritt, which
deals with decomposition of polynomials with complex coefficients
with respect to the operation of composition. In a paper of 1922
Ritt proved out that two maximal decompositions (that is a
decomposition whose components are neither linear nor further
decomposable) of a complex polynomial have the same number of
components and their degrees are the same up to the order. We give a
proof in the spirit of the original paper of Ritt, which uses
concepts like monodromy groups of rational functions, coverings and
theory of blocks in the action of a group on a set.
%Several other proof of this theorem have been given after that,
This result can be applied in the case of an equation involving
compositions of polynomials: thanks to Ritt's theorem we know that
every side of the equation has the same number of indecomposable
component.
\bigskip
%3o capitolo
In the third chapter we give the classical definition of plane
algebraic curves, both in the affine and projective case. We show
that there is a bijection between the points of a non-singular curve
and the valuation rings of its rational function field (which are
called places of the curve). More generally speaking, if we have a
singular curve , the set of valuation rings of its rational
function field is in bijection with the set of points of a
non-singular model of the curve (that is the two curves and
are birational), called desingularization of the curve.
Then we deal with curves whose points are parametrized by a couple
of rational functions in one parameter; we call these curves
rational. From a geometric point of view a rational curve has
desingularization which is a compact Riemann surface of genus zero,
thus isomorphic to . Finally we expose some properties
of parametrizations of rational curves; we show a simple criterium
which provides a necessary and sufficient condition that lets a
rational curve have a polynomial parametrization in terms of places
at infinity.
\bigskip
%4o capitolo
In the fourth chapter we study the aforementioned conjecture of
Schinzel.
For example, if then by taking the full set of
rational numbers we see that the couple satisfies the
Schinzel's property. If is symmetric and the set of rational
points of the curve determined by is infinite, then if we define
to be the projection on the first coordinate of the rational
points of the curve we obtain another example of polynomial with the
above property.
The hypothesis of irreducibility of the polynomial is required
because we want to avoid phenomenon such as and
S=\Q, where is neither linear nor symmetric. In general if a
polynomial has as a factor, then it admits the full
set of rational numbers as set . Another example is the following
(private communication of Schinzel): let
and , where is the Fibonacci sequence
which satisfies the identity for
each natural number ; if f_1,f_2\in\Q[X,Y] are the two
irreducible factors of then for each the couple of
integers is a point of the curve associated to the
polynomial or , according to the parity of .
Zannier has recently given the following counterexample to
Schinzel's conjecture:
with equal to the set of rational (or integer) squares. The idea
is the following: it is well known that for each couple of rational
functions with coefficients in a field
there exists a polynomial such that
. In fact has trascendental degree
one over ; we also say that and are
algebraically dependent. Moreover if we require that the polynomial
is irreducible then it is unique up to multiplication by
constant.
This procedure allows us to build families of polynomials with
Schinzel's property: it is sufficient to take couples of rational
functions , where are
rational functions. If we consider the irreducible polynomial
f\in\Q[X,Y] such that and the set
S=\{\varphi(t)|t\in\Q\}, we see that has Schinzel's
property. In particular Zannier's example is obtained from the
couple of rational functions . If
and then it turns out that is
neither linear nor symmetric in and , but it is a polynomial
with Schinzel's property.
\bigskip
%5o capitolo
In the last chapter we deal with the problem of parametrization of
integer-valued polynomials and we prove the results mentioned at the
beginning of this introduction. The idea of the proof is the
following: let be an integer-valued polynomial as
above; since the set of integer-valued polynomials is a module over
, we can assume that is a prime number . We remark that a
bivariate polynomial of the form has over \Q only two
linear factors; moreover, the set of integer values such that
there exists q\in\Q such that belongs to an irreducible
component of the curve which is not linear in , has
zero density, by a theorem of Siegel. If is
-parametrizable by a polynomial
then by Hilbert's
irreducibility theorem there exists Q\in\Q[\underline{X}] such
that ; we obtain necessary
conditions for such polynomial in order to satisfy the previous
equality. In the same hypothesis, for each there exists
such that . So
we study how the points , for ,
distribute among the irreducible components of the curve
; by the aforementioned theorem of Siegel it turns out
that, up to a subset of density zero of , they belong to
components determined by linear factors of . For each of
them, the projection on the first component of this kind of points
is a set of integers contained in a single residue class modulo the
prime . So if is greater then two, which is the maximum
number of linear factors of a bivariate separated polynomial over
\Q, the set is not -parametrizable.
The problem of factorization of bivariate separated polynomials,
that is polynomials of the form , is a topic which has
been intensively studied for years (Cassels, Fried, Feit, Bilu,
Tichy, Zannier, Avanzi, Cassou-Noguès, Schinzel, etc...)
Our next aim is the classification of the integer-valued polynomials
such that is parametrizable with an integer
coefficient polynomial in more than one variable (for example
). I conjecture that such polynomials (except when
) belong to , where is a prime
different from , is an odd integer coprime with and a
positive integer. I show in my work that if is such a
polynomial, then is -parametrizable.
Moreover we want to study the case of number fields, that is the
parametrization of sets , where is the ring of
integers of a number field and such that
, with polynomials with coefficients in the ring