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Guillermo Matera

Publications and source records attributed to Guillermo Matera.

At least 19 recordsLinked to original sources
arXiv2025

A Kronecker algorithm for locally closed sets over a perfect field

We develop a probabilistic algorithm of Kronecker type for computing a Kronecker representation of a zero-dimensional linear section of an algebraic variety $V$ defined over a perfect field $k$. The variety $V$ is the Zariski closure of the set of common zeros $\{F_1=0,\ldots,F_r=0,G\not=0\}$ of multivariate polynomials $F_1,\ldots,F_r\in k[X_1,\ldots,X_n]$ outside a prescribed hypersurface $\{G=0\}$. We assume that $F_1,\ldots,F_r$ satisfy natural geometric conditions, such as regularity and radicality, in the local ring $k[X_1,\ldots,X_n]_G$. Our approach combines homotopic deformation techniques with symbolic Newton-Hensel lifting and elimination. We discuss the concept of lifting curves as intermediate geometric objects that enable efficient computation. The complexity of the algorithm is expressed in terms of the degrees and arithmetic size of the input and achieves soft-quadratic complexity in these parameters. We provide detailed complexity analyses for arbitrary perfect fields, as well as for two important cases in computer algebra: finite fields and the field of rational numbers. For each case, we obtain sharp bounds on the size of the base field or required primes.

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arXiv2025

Entropy approximations of algebraic matroids over finite fields

We investigate the asymptotic behavior of entropy polymatroids associated with algebraic matroids over finite fields. Given an algebraic matroid ${\sf M}:=(\mathcal{E},r)$ and the irreducible variety $V$ associated with ${\sf M}$, we consider the polymatroid $h_{\mathbb{F}}$ induced by the entropies of the projections of $V(\mathbb{F})$, where $\mathbb{F}$ is a finite extension of $\mathbb{G}$. Revisiting a construction of Mat\'u$\check{\mathrm{s}}$, we show that the polymatroid $h_{\mathbb{F}}$ converges to the rank function $r$ of ${\sf M}$ as $q:=|\mathbb{F}|$ tends to infinity. Our main contribution is to make this convergence quantitative: we derive explicit uniform error bounds for the deviation $|h_\mathbb{F}-r|$, expressed in terms of the degree of $V$, the ground set size $|\mathcal{E}|$, the rank $r(\mathcal{E})$, and $q$. The proofs combine tools of algebraic geometry (effective Lang-Weil estimates and intrinsic degree bounds for annihilating polynomials of circuits) with information-theoretic arguments (submodularity of entropy and conditional entropy estimates). These results provide the first effective and uniform approximation bounds for algebraic matroids by entropy polymatroids, clarifying the quantitative link between algebraic independence (captured by matroid rank) and information-theoretic independence (captured by entropy).

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arXiv2023

Smooth symmetric systems over a finite field and applications

We study the set of common $\mathbb{F}_q$-rational solutions of "smooth" systems of multivariate symmetric polynomials with coefficients in a finite field $\mathbb{F}_q$. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of $\mathbb{F}_q$ has a "good" geometric behavior. This allows us to obtain precise estimates on the corresponding number of common $\mathbb{F}_q$-rational solutions. In the case of hypersurfaces we are able to improve the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.

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arXiv2022

The distribution of defective multivariate polynomial systems over a finite field

This paper deals with properties of the algebraic variety defined as the set of zeros of a "deficient" sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely irreducible varieties. For these types, we establish improved bounds on the dimension of the set of deficient systems of each type over an arbitrary field. On the other hand, we establish improved upper bounds on the number of systems of each type over a finite field.

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arXiv2022

On the computation of rational solutions of underdetermined systems over a finite field

We design and analyze an algorithm for computing solutions with coefficients in a finite field $\mathbb{F}_q$ of underdetermined systems defined over $\mathbb{F}_q$. The algorithm is based on reductions to zero-dimensional searches. The searches are performed on "vertical strips", namely parallel linear spaces of suitable dimension in a given direction. Our results show that, on average, less than three searches suffice to obtain a solution of the original system, with a probability of success which grows exponentially with the number of searches. The analysis of our algorithm relies on results on the probability that the solution set (over the algebraic closure of $\mathbb{F}_q$) of a random system with coefficients in $\mathbb{F}_q$ satisfies certain geometric and algebraic properties which is of independent interest.

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arXiv2021

Interpolation by decomposable univariate polynomials

The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e, respectively, with de=n, and therefore d+e-1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists.

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arXiv2020

Average-case complexity of the Euclidean algorithm with a fixed polynomial over a finite field

We analyze the behavior of the Euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field F_q of q elements when the highest-degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal bounds in terms of q for the number of elements f which are relatively prime with g and for the average degree of gcd(g,f). The accuracy of our estimates is confirmed by practical experiments. We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs (g,f) as above.

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arXiv2019

Shifted varieties and discrete neighborhoods around varieties

In the area of symbolic-numerical computation within computer algebra, an interesting question is how "close" a random input is to the "critical" ones, like the singular matrices in linear algebra or the polynomials with multiple roots for Newton's root-finding method. Bounds, sometimes very precise, are known for the volumes over R or C of such neighborhoods of the varieties of "critical" inputs. This paper deals with the discrete version of this question: over a finite field, how many points lie in a certain type of neighborhood around a given variety? A trivial upper bound on this number is (size of the variety) x (size of a neighborhood of a point). It turns out that this bound is usually asymptotically tight, including for the singular matrices, polynomials with multiple roots, and pairs of non-coprime polynomials. The interesting question then is: for which varieties does this bound not hold? We show that these are precisely those that admit a shift, that is, where one absolutely irreducible component is a shift (translation by a fixed nonzero point) of another such component. Furthermore, the shift-invariant absolutely irreducible varieties are characterized as being cylinders over some base variety. Computationally, determining whether a given variety is shift-invariant turns out to be intractable, namely NP-hard even in simple cases.

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arXiv2018

Factorization patterns on nonlinear families of univariate polynomials over a finite field

We estimate the number $|\mathcal{A}_{\boldsymbolλ}|$ of elements on a nonlinear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $r$ having factorization pattern $\boldsymbolλ:=1^{λ_1}2^{λ_2}\cdots r^{λ_r}$. We show that $|\mathcal{A}_{\boldsymbolλ}|= \mathcal{T}(\boldsymbolλ)\,q^{r-m}+\mathcal{O}(q^{r-m-{1}/{2}})$, where $\mathcal{T}(\boldsymbolλ)$ is the proportion of elements of the symmetric group of $r$ elements with cycle pattern $\boldsymbolλ$ and $m$ is the codimension of $\mathcal{A}$. We provide explicit upper bounds for the constants underlying the $\mathcal{O}$--notation in terms of $\boldsymbolλ$ and $\mathcal{A}$ with "good" behavior. We also apply these results to analyze the average--case complexity of the classical factorization algorithm restricted to $\mathcal{A}$, showing that it behaves as good as in the general case.

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arXiv2016

On the bit complexity of polynomial system solving

We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. Our algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a "lucky" prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze.

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arXiv2016

On the computation of rational points of a hypersurface over a finite field

We design and analyze an algorithm for computing rational points of hypersurfaces defined over a finite field based on searches on "vertical strips", namely searches on parallel lines in a given direction. Our results show that, on average, less than two searches suffice to obtain a rational point. We also analyze the probability distribution of outputs, using the notion of Shannon entropy, and prove that the algorithm is somewhat close to any "ideal" equidistributed algorithm.

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arXiv2015

Explicit estimates for polynomial systems defining irreducible smooth complete intersections

This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero "obstruction" polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.

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arXiv2015

Quiz Games as a model for Information Hiding

We present a general computation model inspired in the notion of information hiding in software engineering. This model has the form of a game which we call quiz game. It allows in a uniform way to prove exponential lower bounds for several complexity problems of elimination theory.

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arXiv2015

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

We estimate the average cardinality $\mathcal{V}(\mathcal{A})$ of the value set of a general family $\mathcal{A}$ of monic univariate polynomials of degree $d$ with coefficients in the finite field $\mathbb{F}_{\hskip-0.7mm q}$. We establish conditions on the family $\mathcal{A}$ under which $\mathcal{V}(\mathcal{A})=μ_d\,q+\mathcal{O}(q^{1/2})$, where $μ_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. The result holds without any restriction on the characteristic of $\mathbb{F}_{\hskip-0.7mm q}$ and provides an explicit expression for the constant underlying the $\mathcal{O}$--notation in terms of $d$. We reduce the question to estimating the number of $\mathbb{F}_{\hskip-0.7mm q}$--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over $\mathbb{F}_{\hskip-0.7mm q}$. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of $\mathbb{F}_{\hskip-0.7mm q}$--rational points.

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arXiv2015

Number of rational points of symmetric complete intersections over a finite field and applications

We study the set of common F_q-rational zeros of systems of multivariate symmetric polynomials with coefficients in a finite field F_q. We establish certain properties on these polynomials which imply that the corresponding set of zeros over the algebraic closure of F_q is a complete intersection with "good" behavior at infinity, whose singular locus has a codimension at least two or three. These results are used to estimate the number of F_q-rational points of the corresponding complete intersections. Finally, we illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.

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arXiv2014

Explicit Estimates for the Number of Rational Points of Singular Complete Intersections over a Finite Field

Let $V\subset\mathbb{P}^n(\overline{F}_{\hskip-0.7mm q})$ be a complete intersection defined over a finite field $F_{\hskip-0.7mm q}$ of dimension $r$ and singular locus of dimension at most $0\le s\le r-2$. We obtain an explicit version of the Hooley--Katz estimate $||V(F_{\hskip-0.7mm q})|-p_r|=\mathcal{O}(q^{(r+s+1)/2})$, where $|V(F_{\hskip-0.7mm q})|$ denotes the number of $F_{\hskip-0.7mm q}$-rational points of $V$ and $p_r:=|\mathbb{P}^r(F_{\hskip-0.7mm q})|$. Our estimate improves all the previous estimates in several important cases. Our approach relies on tools of classical algebraic geometry. A crucial ingredient is a new effective version of the Bertini smoothness theorem, namely an explicit upper bound of the degree of a proper Zariski closed subset of $(\mathbb P^{n})^{s+1}(\overline{F}_{\hskip-0.7mm q})$ which contains all the singular linear sections of $V$ of codimension $s+1$.

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arXiv2014

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

We obtain estimates on the number $|\mathcal{A}_{\boldsymbolλ}|$ of elements on a linear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $n$ having factorization pattern $\boldsymbolλ:=1^{λ_1}2^{λ_2}\cdots n^{λ_n}$. We show that $|\mathcal{A}_{\boldsymbolλ}|= \mathcal{T}(\boldsymbolλ)\,q^{n-m}+\mathcal{O}(q^{n-m-{1}/{2}})$, where $\mathcal{T}(\boldsymbolλ)$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $\boldsymbolλ$ and $m$ is the codimension of $\mathcal{A}$. Furthermore, if the family $\mathcal{A}$ under consideration is "sparse", then $|\mathcal{A}_{\boldsymbolλ}|= \mathcal{T}(\boldsymbolλ)\,q^{n-m}+\mathcal{O}(q^{n-m-{1}})$. 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 $\boldsymbolλ$ 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.

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