arXiv subjects

Luis Miguel Pardo

Publications and source records attributed to Luis Miguel Pardo.

7 recordsLinked to original sources

arXiv2026

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.

math.AG↗

arXiv2025

Erzeugunsgrad, VC-Dimension and Neural Networks with rational activation function

The notion of Erzeugungsgrad was introduced by Joos Heintz in 1983 to bound the number of non-empty cells occurring after a process of quantifier elimination. We extend this notion and the combinatorial bounds of Theorem 2 in Heintz (1983) using the degree for constructible sets defined in Pardo-Sebastián (2022). We show that the Erzeugungsgrad is the key ingredient to connect affine Intersection Theory over algebraically closed fields and the VC-Theory of Computational Learning Theory for families of classifiers given by parameterized families of constructible sets. In particular, we prove that the VC-dimension and the Krull dimension are linearly related up to logarithmic factors based on Intersection Theory. Using this relation, we study the density of correct test sequences in evasive varieties. We apply these ideas to analyze parameterized families of neural networks with rational activation function.

cs.LG↗

arXiv2022

An unfeasability view of neural network learning

We define the notion of a continuously differentiable perfect learning algorithm for multilayer neural network architectures and show that such algorithms don't exist provided that the length of the data set exceeds the number of involved parameters and the activation functions are logistic, tanh or sin.

cs.LG↗

arXiv2021

A promenade through Correct Test Sequences I: Degree of constructible sets, Bézout's Inequality and density

In Heintz-Schnorr (1982), the authors introduced the notion of correct test sequence and since then it has been widely used to design probabilistic algorithms for Polynomial Equality Test. The aim of this manuscript is to study the foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent and appear in many different forms in the mathematical literature: As identity sequences for Function Identity Test, as norming sets in the field of Banach algebras or as samples in the context of Reproducing Kernel Hilbert Spaces. As main outcome, we generalize the main statement of Heintz-Schnorr (1982) proving that short correct test sequences for constructible sets of lists of polynomials do exist and are densely distributed in any constructible set of accurate dimension and degree. The main tool used to prove this result is the theory of degree of constructible sets, which we introduce and develop in this manuscript, generalizing the results of Heintz (1983) and proving two Bezout's Inequalities for two different notions of degree. We present a ${\bf BPP}_K$ algorithm to exhibit the power of correct test sequences, this algorithm decides whether a list of polynomials is a secant sequence by just evaluating the input list at some well-suited points. We show the differences between correct test sequences and Demillo-Lipton-Schwartz-Zippel probabilistic tests and we reformulate, prove and generalize two well-known results of the Polynomial Method: We prove Dvir's exponential lower bounds for Kakeya sets from lower bounds for the length of correct test sequences and generalize Alon's Combinatorial Nullstellensatz.

math.AG↗

arXiv2017

On Bezout Inequalities for non-homogeneous Polynomial Ideals

We introduce a "workable" notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic Bézout Inequalities for the sum of two ideals in terms of this notion of degree and the degree of generators. We compute probabilistically the degree of an equidimensional ideal.

cs.SC↗

arXiv2003

The hardness of polynomial equation solving

In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids "unnecessary" branchings and that P admits the efficient computation of certain natural limit objects (as e.g. the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.

math.AC↗

arXiv1999

Sharp estimates for the arithmetic Nullstellensatz

We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over $\Z$. The result improves previous work of Philippon, Berenstein-Yger and Krick-Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application, we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine variety defined over a number field. We introduce this notion and study its basic properties.

math.AG↗