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Carlos D'Andrea

Publications and source records attributed to Carlos D'Andrea.

At least 19 recordsLinked to original sources
arXiv2026

Toric Euler-Jacobi vanishing theorem and zeros at infinity

Residues appear naturally in various questions in complex and algebraic geometry: interpolation, duality, representation problems, and obstructions. The first global vanishing result in the projective plane, known as the Euler-Jacobi theorem, was established by Jacobi in 1835. In the toric case, the input is a system of n Laurent sparse polynomials with fixed Newton polytopes, and the first version of the Euler-Jacobi toric vanishing theorem for residues in the n-torus is due to Khovanskii in 1978, under restrictive genericity assumptions. In this paper, we provide geometric conditions on the input Newton polytopes to ensure that this global vanishing is equivalent to the existence of zeros at infinity in the associated compact toric variety. We relate these conditions to the dimension at the toric critical degree of the quotient of the Cox ring by the ideal generated by the (multi)homogenizations of the input polynomials. We also relate the existence of zeros at infinity to interpolation questions.

math.AG↗
arXiv2025

On the effective Pourchet's Theorem

With the aid of Hensel Lemma, we refine the 2-adic Newton polygon algorithm proposed by Magron, Koprowski, and Vaccon at ISSAC 2023 to express computationally a given positive univariate polynomial with rational coefficients as a sum of five squares of rational polynomials -the effective Pourchet's Theorem- and extend it to cover almost all the possible inputs. We also provide examples which are covered with our methods but cannot be detected by previous conjectural algorithms.

math.NT↗
arXiv2025

Sparse Nullstellensatz, resultants and determinants of complexes

We refine and extend a result by Tuitman on the supports of a Bezout identity satisfied by a finite sequence of sparse Laurent polynomials without common zeroes in the toric variety associated to their supports. When the number of these polynomials is one more than the dimension of the ambient space, we obtain a formula for computing the sparse resultant as the determinant of a Koszul type complex.

math.AG↗
arXiv2024

Entropy numbers and box dimension of polynomials and holomorphic functions

We study entropy numbers and box dimension of (the image of) homogeneous polynomials and holomorphic functions between Banach spaces. First, we see that entropy numbers and box dimensions of subsets of Banach spaces are related. We show that the box dimension of the image of a ball under a homogeneous polynomial is finite if and only if it spans a finite-dimensional subspace, but this is not true for holomorphic functions. Furthermore, we relate the entropy numbers of a holomorphic function to those of the polynomials of its Taylor series expansion. As a consequence, if the box dimension of the image of a ball by a holomorphic function $f$ is finite, then the entropy numbers of the polynomials in the Taylor series expansion of $f$ at any point of the ball belong to $\ell_p$ for every $p>1$.

math.FA↗
arXiv2023

Subresultants and the Shape Lemma

In nice cases, a zero-dimensional complete intersection ideal over a field of characteristic zero has a Shape Lemma. There are also cases where the ideal is generated by the resultant and first subresultant polynomials of the generators. This paper explores the relation between these representations and studies when the resultant generates the elimination ideal. We also prove a Poisson formula for resultants arising from the hidden variable method.

math.AC↗
arXiv2021

The Canny-Emiris conjecture for the sparse resultant

We present a product formula for the initial parts of the sparse resultant associated to an arbitrary family of supports, generalising a previous result by Sturmfels. This allows to compute the homogeneities and degrees of the sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain a similar product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated to a mixed subdivision of a polytope. Applying these results, we prove that the sparse resultant can be computed as the quotient of the determinant of such a square matrix by a certain principal minor, under suitable hypothesis. This generalises the classical Macaulay formula for the homogeneous resultant, and confirms a conjecture of Canny and Emiris.

math.AC↗
arXiv2021

Bounds for degrees of syzygies of polynomials defining a grade two ideal

We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.

math.AC↗
arXiv2020

Degree and birationality of multi-graded rational maps

We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the "saturated special fiber ring", which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.

math.AG↗
arXiv2019

Minimal solutions of the rational interpolation problem

We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies with the standard method provided by the Extended Euclidean Algorithm. As a consequence, we obtain explicit descriptions for solutions of "minimal" degrees in terms of the degrees of elements appearing in the EEA. This allows us to describe the minimal degree in a $μ$-basis of a polynomial planar parametrization in terms of a "critical" degree arising in the EEA.

math.AC↗
arXiv2019

The geometry of the flex locus of a hypersurface

We give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces. Using this formula, we compute the dimension of this flex locus, and an upper bound for the degree of its defining equations. We also show that, when the hypersurface is generic, this bound is reached, and that the generic flex line is unique and has the expected order of contact with the hypersurface.

math.AG↗
arXiv2019

The Rees Algebra of Parametric Curves via liftings

We study the defining equations of the Rees algebra of ideals arising from curve parametrizations in the plane and in rational normal scrolls, inspired by the work of Madsen and Kustin, Polini and Ulrich. The curves are related by work of Bernardi, Gimigliano and Ida, and we use this framework to relate the defining equations.

math.AC↗
arXiv2018

Closed formula for univariate subresultants in multiple roots

We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is the first closed formula for subresultants in terms of roots that works for arbitrary polynomials, previous efforts only handled special cases. Our extension involves in some cases confluent Schur polynomials, and is obtained by using multivariate symmetric interpolation via an Exchange Lemma.

math.AC↗
arXiv2017

Quantitative equidistribution of Galois orbits of small points in the N-dimensional torus

We present a quantitative version of Bilu's theorem on the limit distribution of Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus. Our result gives, for a given point, an explicit bound for the discrepancy between its Galois orbit and the uniform distribution on the compact subtorus, in terms of the height and the generalized degree of the point.

math.NT↗
arXiv2017

Reductions Modulo Primes of Systems of Polynomial Equations and Algebraic Dynamical Systems

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the algebraic closure of the field with $p$ elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture.

math.NT↗
arXiv2017

Subresultants in multiple roots: an extremal case

We provide explicit formulae for the coefficients of the order-d polynomial subresultant of (x-α)^m and (x-β)^n with respect to the set of Bernstein polynomials \{(x-α)^j(x-β)^{d-j}, \, 0\le j\le d\}. They are given by hypergeometric expressions arising from determinants of binomial Hankel matrices.

math.AC↗
arXiv2017

Orbits of Polynomial Dynamical Systems Modulo Primes

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly improve a result of Silverman~(2008) and recover a result of Akbary and Ghioca~(2009) as special extreme cases of our estimates.

math.NT↗
arXiv2014

A Poisson formula for the sparse resultant

We present a Poisson formula for sparse resultants and a formula for the product of the roots of a family of Laurent polynomials, which are valid for arbitrary families of supports. To obtain these formulae, we show that the sparse resultant associated to a family of supports can be identified with the resultant of a suitable multiprojective toric cycle in the sense of Remond. This connection allows to study sparse resultants using multiprojective elimination theory and intersection theory of toric varieties.

math.AG↗