arXiv subjects

Melina Privitelli

Publications and source records attributed to Melina Privitelli.

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

Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields

Let $N$ denote the number of solutions to the generalized Markoff-Hurwitz-type equation \[(a_1X_1^m+\cdots + a_nX_n^m+a)^k=bX_1\cdots X_n \] over the finite field $\mathbb{F}_q$, where $m,k$ are positive integers, and $a,b,a_i\in \mathbb{F}_q^*$ for $i=1,\dots, n$, with $k,m\ge 2$ and $n\ge 3$. Using techniques from algebraic geometry, we provide an estimate for $N$ and establish conditions under which the equation admits solutions where all $X_i$ are nonzero.

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arXiv2024

Singularly cospectral circulant graphs

Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse may be false. In this paper, we present sufficient conditions for two circulant graphs, with an even number of vertices, to be noncospectral singularly cospectral graphs. In this analysis, we study when a pair of these graphs have the same or distinct inertia. In addition, we show that two singularly cospectral circulant graphs with an odd prime number of vertices are isomorphic.

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arXiv2024

An approach to the moments subset sum problem through systems of diagonal equations over finite fields

Let $\mathbb{F}_q$ be the finite field of $q$ elements, for a given subset $D\subset \mathbb{F}_q$, $m\in \mathbb{N}$, an integer $k\leq |D|$ and $\boldsymbol{b}\in \mathbb{F}_q^m$ we are interested in determining the existence of a subset $S\subset D$ of cardinality $k$ such that $\sum_{a\in S}a^i=b_i$ for $i=1,\ldots, m$. This problem is known as the moment subset sum problem and it is $NP$-complete for a general $D$. We make a novel approach of this problem trough algebraic geometry tools analyzing the underlying variety and employing combinatorial techniques to estimate the number of $\mathbb{F}_q$-rational points on certain varieties. We managed to give estimates on the number of $\mathbb{F}_q$-rational points on certain diagonal equations and use this results to give estimations and existence results for the subset sum problem.

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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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arXiv2020

Playing through a noisy channel (and knowing it)

In this note we discuss a theory of combinatorial games that involve transmitting the moves through a noisy channel that can introduce errors during the transmission. Players are aware of this interference and incorporate this variable into the game: the valid move is the received one, regardless of whether it is the other player's sent move (as long as it is a valid move in the original game; otherwise, a retransmission is requested). Players know the probability of introducing an error through communication and can play a non-optimal (but valid) move that maximizes their chances of winning. We present some examples and provide the basic definitions and results of this type of games.

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arXiv2020

On the number of simultaneous solutions of certain diagonal equations over finite fields

In this paper we obtain explicit estimates and existence results on the number of $\mathbb{F}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.

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arXiv2020

Estimates on the number of rational solutions of variants of diagonal equations over finite fields

In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of the set of common zeros of symmetric polynomials over the algebraic closure of a finite field. We provide improved estimates and existence results of rational solutions to the following equations: deformed diagonal equations, generalized Markoff Hurwitz type equations and Carlitz's equations. We extend these techniques to a more general variants of diagonal equations over finite fields.

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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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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 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

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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arXiv2013

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

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},..., a_{d-s} are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d,s,\bfs{a})=μ_d.q+\mathcal{O}(1), where V(d,s,\bfs{a}) is such an average cardinality, μ_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the \mathcal{O}--notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq--rational points is established.

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arXiv2013

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

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},...,a_{d-s} are fixed. Our estimate asserts that \mathcal{V}(d,s,\bfs{a})=μ_d\,q+\mathcal{O}(q^{1/2}), where \mathcal{V}(d,s,\bfs{a}) is such an average cardinality, μ_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},...,a_{d-s}). We also prove that \mathcal{V}_2(d,s,\bfs{a})=μ_d^2\,q^2+\mathcal{O}(q^{3/2}), where that \mathcal{V}_2(d,s,\bfs{a}) is the average second moment on any family of monic polynomials of Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that \mathcal{V}_2(d,0)=μ_d^2\,q^2+\mathcal{O}(q), where \mathcal{V}_2(d,0) denotes the average second moment of all monic polynomials in Fq[T] of degree d with f(0)=0. All our estimates hold for fields of characteristic p>2 and provide explicit upper bounds for the constants underlying the \mathcal{O}--notation in terms of d and s with "good" behavior. Our approach reduces the questions to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. A critical point for our results is an analysis of the singular locus of the varieties under consideration, which allows to obtain rather precise estimates on the corresponding number of Fq--rational points.

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