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Carlos D'Andrea

Publications and source records attributed to Carlos D'Andrea.

At least 37 records · Page 2Linked to original sources
arXiv2014

The Rees Algebra of a monomial plane parametrization

We compute a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve. We describe explicitly both the bigraded Betti numbers and the maps of the resolution in terms of a generalized version of the Euclidean Algorithm. We also explore the relation between pencils of adjoints of the monomial plane curve and elements in a suitable piece of the defining ideal of the Rees Algebra.

math.AC↗
arXiv2014

Quantitative equidistribution for the solutions of systems of sparse polynomial equations

For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},..., x_n^{\pm1}] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of the system of equations f_1=\dots=f_n=0, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erdos and Turan on the distribution of the arguments of the roots of a univariate polynomial. We apply this result to bound the number of real roots of a system of Laurent polynomials, and to study the asymptotic distribution of the roots of systems of Laurent polynomials with integer coefficients, and of random systems of Laurent polynomials with complex coefficients.

math.CV↗
arXiv2014

Subresultants, Sylvester sums and the rational interpolation problem

We present a solution for the classical univariate rational interpolation problem by means of (univariate) subresultants. In the case of Cauchy interpolation (interpolation without multiplicities), we give explicit formulas for the solution in terms of symmetric functions of the input data, generalizing the well-known formulas for Lagrange interpolation. In the case of the osculatory rational interpolation (interpolation with multiplicities), we give determinantal expressions in terms of the input data, making explicit some matrix formulations that can independently be derived from previous results by Beckermann and Labahn.

math.AC↗
arXiv2014

Moving curve ideals of rational plane parametrizations

In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the Computer Aided Geometric Design Community. The analysis of the validity of these techniques has been a fruitful ground for Commutative Algebraists and Algebraic Geometers, and several results have been obtained so far. Yet, a lot of research is still being done currently around this topic. In this note we present these methods, show their mathematical formulation, and survey current results and open questions.

math.AC↗
arXiv2012

Subresultants in Multiple Roots

We extend our previous work on Poisson-like formulas for subresultants in roots to the case of polynomials with multiple roots in both the univariate and multivariate case, and also explore some closed formulas in roots for univariate polynomials in this multiple roots setting.

math.AC↗
arXiv2012

Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

We present bounds for the degree and the height of the polynomials arising in some central problems in effective algebraic geometry including the implicitation of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.

math.AG↗
arXiv2012

Rational plane curves parameterizable by conics

We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable quadratic transformations in projective plane. We also describe all the possible proper parameterizations of them, and a set of minimal generators of the Rees Algebra associated to these parameterizations, extending well-known results for curves parameterizable by lines.

math.AG↗
arXiv2012

Singular factors of rational plane curves

We give a complete factorization of the invariant factors of resultant matrices built from birational parameterizations of rational plane curves in terms of the singular points of the curve and their multiplicity graph. This allows us to prove the validity of some conjectures about these invariants stated by Chen, Wang and Liu in [J. Symbolic Comput. 43(2):92-117, 2008]. As a byproduct, we also give a complete factorization of the D-resultant for rational functions in terms of the similar data extracted from the multiplicities.

math.AC↗
arXiv2010

Minimal generators of the defining ideal of the Rees Algebra associated to monoid parametrizations

We describe a minimal set of generators of the defining ideal of the Rees Algebra associated to a proper parametrization of any monoid hypersurface. In the case of plane curves, we recover a known description for rational parametrizations having a syzygy of minimal degree (μ=1). We also show that our approach can be applied to parametrizations of rational surfaces having a Hilbert-Burch resolution with μ_1=μ_2=1.

math.AC↗
arXiv2010

The Newton polygon of a rational plane curve

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kushnirenko-Bernstein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational functions, with explicit genericity conditions. We also show that the variety of rational curves with given Newton polygon is unirational and we compute its dimension. As a consequence, we obtain that any convex lattice polygon with positive area is the Newton polygon of a rational plane curve.

math.AG↗
arXiv2009

Tropicalization and irreducibility of Generalized Vandermonde Determinants

We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determinants whose tropicalization with respect to the variables of a row is irreducible.

math.AC↗
arXiv2008

Rational Formulas for Traces in zero-dimensional Algebras

We present a rational expression for the trace of the multiplication map M_r in a finite-dimensional algebra of the form A:=K[x_1,...,x_n]/I in terms of the generalized Chow form of I. Here, I is a zero-dimensional ideal of K[x_1,...,x_n] is a zero-dimensional ideal, K is a field of characteristic zero, and r(x_1,..., x_n) a rational function whose denominator is not a zero divisor in A. If I is a complete intersection in the torus, we get numerator and denominator formulas for traces in terms of sparse resultants.

math.AC↗
arXiv2008

Sylvester's Double Sums: the general case

In 1853 Sylvester introduced a family of double sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to charactise all members of the family, providing a uniform method.

math.AC↗
arXiv2006

An Elementary Proof of Sylvester's Double Sums for Subresultants

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz by using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.

math.AC↗
arXiv2006

On the Mahler measure of resultants in small dimensions

We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then compute the Mahler measure of resultants in dimension two, and examples in dimension three and four. Finally, we show that sparse resultants are tempered polynomials. This property suggests that their Mahler measure may lead to special values of L-functions and polylogarithms.

math.NT↗
arXiv2006

On the Jacobian ideal of the binary discriminant

Let $Δ$ denote the discriminant of a generic binary $d$-ic. We show that for $d \ge 3$, the Jacobian ideal of $Δ$ is perfect of height 2. Moreover, we describe its SL_2-equivariant minimal resolution, and the associated invariant differential equations satisfied by $Δ$. A similar result is proved for the resultant of two forms of orders $d,e$, whenever $d \ge e-1$. We also explain the role of the Morley form in the determinantal formula for the resultant; this relies upon a calculation which is done in the appendix by A. Abdesselam.

math.AG↗