Macaulay style formulas for sparse resultants

We present formulas for computing the resultant of sparse polyno- mials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials

Ideals de corbes mòbils i la seva interacció amb el disseny assistit per ordinador

Presentem un cas d'interacció fructífera entre l'àlgebra commutativa i el disseny assistit per ordinador. Els problemes en aquesta àrea aplicada i cada vegada més important de la informàtica s'han traslladat a l'estudi d'estructures algebraiques abstractes, i han enriquit la matemàtica amb múltiples resultats teòrics i problemes oberts que expliquem en aquest text

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

Sobre Corbes paramètriques i polígons de Newton

Les corbes i superfícies algebraiques poden ser definides implícitament com a solucions d'equacions polinomials i, de vegades, també poden definir-se paramètricament, mitjançant funcions racionals. Plantegem el problema de la conversió d'una d'aquestes formes de representació a l'altra. A continuació, explorem la possibilitat d'obtenir, a partir de les equacions paramètriques i sense necessitat d'efectuar l'operació costosa de la implicitació, un objecte pròxim a les equacions implícites associades: el polítop de Newton d'una hipersuperfície donada paramètricament

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 in $\mathbb{P}^{3}$. 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

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

The set of unattainable points for the Rational Hermite Interpolation Problem

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all these objects

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 slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates

On the Jacobian ideal of the binary discriminant (with an appendix by Abdelmalek Abdesselam)

Let ∆ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of ∆ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by ∆. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e − 1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d − n, then we show that the ideal of Φn is also perfect, and we construct a covariant which characterizes this locus. 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

Macaulay style formulas for sparse resultants

We present formulas for computing the resultant of sparse polyno- mials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials