Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Let P^n denote the n-dimensional projective space defined over the algebraic closure of a finite field F_q, let V contained P^n be a complete intersection defined over F_q of dimension r and singular locus of dimension at most s, and let \pi:V-->P^{s+1} be a "generic" linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning \pi, namely an explicit upper bound of the degree of a proper Zariski closed subset of P^{s+1} which contains all the points defining singular fibers of \pi. For this purpose we make essential use of the concept of polar variety associated to the set of exceptional points of \pi. As a consequence of our effective Bertini theorem we obtain results of existence of smooth rational points of V, namely conditions on q which imply that V has a smooth q-rational point. Finally, for s=r-2 and s=r-3 we obtain estimates on the number of q-rational points of V, and we discuss how these estimates can be used in order to determine the average value set of "small" families of univariate polynomials with coefficients in F_q.

Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

Explicit estimates for the number of rational points of singular complete intersections over a finite field

We obtain estimates on the number $|\mathcal{A}_{\boldsymbol{\lambda}}|$ of elements on a linear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $n$ having factorization pattern $\boldsymbol{\lambda}:=1^{\lambda_1}2^{\lambda_2}\cdots n^{\lambda_n}$. We show that $|\mathcal{A}_{\boldsymbol{\lambda}}|= \mathcal{T}(\boldsymbol{\lambda})\,q^{n-m}+\mathcal{O}(q^{n-m-{1}/{2}})$, where $\mathcal{T}(\boldsymbol{\lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $\boldsymbol{\lambda}$ and $m$ is the codimension of $\mathcal{A}$. Furthermore, if the family $\mathcal{A}$ under consideration is "sparse", then $|\mathcal{A}_{\boldsymbol{\lambda}}|= \mathcal{T}(\boldsymbol{\lambda})\,q^{n-m}+\mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $\mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $\mathcal{O}$--notation in terms of $\boldsymbol{\lambda}$ and $\mathcal{A}$ with "good" behavior. Our approach reduces the question to estimate the number of $\mathbb{F}_q$--rational points of certain families of complete intersections defined over $\mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $\mathbb{F}_q$--rational points are established.

The distribution of factorization patterns on linear families of polynomials over a finite field

This paper describes the probabilistic behaviour of a random Sturmian word. It performs the probabilistic analysis of the recurrence function which can be viewed as a waiting time to discover all the factors of length $n$ of the Sturmian word. This parameter is central to combinatorics of words. Having fixed a possible length $n$ for the factors, we let $\alpha$ to be drawn uniformly from the unit interval $[0,1]$, thus defining a random Sturmian word of slope $\alpha$. Thus the waiting time for these factors becomes a random variable, for which we study the limit distribution and the limit density.

The recurrence function of a random Sturmian word

On the value set of small families of polynomials over a finite field, I

Fil: Cafure, Antonio Artemio. Universidad de Buenos Aires. Ciclo Basico Comun; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

https://www.jstor.org/stable/10.4169/amer.math.monthly.124.1.37

Irreducibility criteria for reciprocal polynomials and applications

This paper is a first attempt to describe the probabilistic behaviour of a random Sturmian word. It performs the probabilistic analysis of the recurrence function which provides precise information on the structure of such a word. With each Sturmian word of slope $\alpha $, we associate particular sequences of factor lengths which have a given “position” with respect to the sequence of continuants of $\alpha $, we then let $\alpha $ to be uniformly drawn inside the unit interval [0,1]. This probabilistic model is well-adapted to better understand the role of the position in the recurrence properties.

Recurrence Function on Sturmian Words: A Probabilistic Study

This paper has been withdrawn 
Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$. Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$. By imposing $n_{_N}\le m \forall N\in \zN$, Jarnik defined the corresponding sets $E_m$ and gave a first estimate of $d_H(E_m)$, $d_H$ the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with $d_H(E_m)$ and $d_H(F_m)$, $F_m$ being the set of real numbers for which ${\sum_{i=1}^N n_i\over N}\le m$.

On the Hausdorff dimension of certain sets arising in Number Theory

This article is devoted to the study of two arithmetical sources associated with classical partitions, that are both defined through the mediant of two fractions The Stern-Brocot source is associated with the sequence of all the mediants, while the Sturm source only keeps mediants whose denominator is "not too large" Even though these sources are both of zero Shannon entropy, with very similar Renyi entropies, their probabilistic features yet appear to be quite different We then study how they influence the behaviour of tries built on words they emit, and we notably focus on the trie depth The paper deals with Analytic Combinatorics methods, and Dirichlet generating functions, that are usually used and studied in the case of good sources with positive entropy To the best of our knowledge, the present study is the first one where these powerful methods are applied to a zero-entropy context In our context, the generating function associated with each source is explicit and related to classical functions in Number Theory, as the ζ function, the double ζ function or the transfer operator associated with the Gauss map We obtain precise asymptotic estimates for the mean value of the trie depth that prove moreover to be quite different for each source Then, these sources provide explicit and natural instances which lead to two unusual and different trie behaviours

https://hal.archives-ouvertes.fr/hal-02868527/document

Eda Cesaratto

Papers

On the value set of small families of polynomials over a finite field, I

Irreducibility criteria for reciprocal polynomials and applications

Recurrence Function on Sturmian Words: A Probabilistic Study

On the Hausdorff dimension of certain sets arising in Number Theory

Two Arithmetical Sources and Their Associated Tries