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Laurent Busé

Publications and source records attributed to Laurent Busé.

29 records · Page 2Linked to original sources
arXiv2016

Effective criteria for bigraded birational maps

In this paper, we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that became to be known as Jacobian dual matrices. Then, we focus on rational maps from the product of two projectine lines to the projective plane in very low bidegrees and provide new matrix-based birationality criteria by analyzing the syzygies of the defining equations of the map, in particular by looking at the dimension of certain bigraded parts of the syzygy module. Finally, applications of our results to the context of geometric modeling are discussed at the end of the paper.

math.AC↗
arXiv2014

Resultant of an equivariant polynomial system with respect to the symmetric group

Given a system of n homogeneous polynomials in n variables which is equivariant with respect to the canonical actions of the symmetric group of n symbols on the variables and on the polynomials, it is proved that its resultant can be decomposed into a product of several smaller resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial.

math.AC↗
arXiv2013

Fitting ideals and multiple-points of surface parameterizations

Given a birational parameterization of an algebraic surface S in the projective space, the purpose of this paper is to investigate the sets of points on S whose preimage consists in k or more points, counting multiplicities. They are described explicitly in terms of Fitting ideals of some graded parts of the symmetric algebra associated to this parameterization.

math.AC↗
arXiv2012

A computational approach to the discriminant of homogeneous polynomials

In this paper, the discriminant of homogeneous polynomials is studied in two particular cases: a single homogeneous polynomial and a collection of n-1 homogeneous polynomials in n variables. In these two cases, the discriminant is defined over a large class of coefficient rings by means of the resultant. Many formal properties and computational rules are provided and the geometric interpretation of the discriminant is investigated over a general coefficient ring, typically a domain.

math.AC↗
arXiv2011

On the total order of reducibility of a pencil of algebraic plane curves

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d^2-1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newton's polygon of the pencil.

math.AC↗
arXiv2009

Elimination and nonlinear equations of Rees algebra

A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -- which of course corresponds to non linear equations of the Rees algebra -- can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, half of the degrees where the torsion does not vanish are understood.

math.AC↗
arXiv2009

Noether's forms for the study of non-composite rational functions and their spectrum

In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r. With the same strategy, we also study the decomposability of r modulo p. Some similar explicit results are also provided for the case of polynomials with coefficients in a polynomial ring.

math.NT↗
arXiv2009

On the equations of the moving curve ideal of a rational algebraic plane curve

Given a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence.

math.AG↗
arXiv2007

Explicit factors of some iterated resultants and discriminants

In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into irreducible factors of several constructions involving two times iterated univariate resultants and discriminants over the integer universal ring of coefficients of the entry polynomials. Cases involving from two to four generic polynomials and resultants or discriminants in one of their variables are treated. The decompositions into irreducible factors we get are obtained by exploiting fundamental properties of the univariate resultants and discriminants and induction on the degree of the polynomials. As a consequence, each irreducible factor can be separately and explicitly computed in terms of a certain multivariate resultant. With this approach, we also obtain as direct corollaries some results conjectured by Collins and McCallum which correspond to the case of polynomials whose coefficients are themselves generic polynomials in other variables. Finally, a geometric interpretation of the algebraic factorization of the iterated discriminant of a single polynomial is detailled.

cs.SC↗
arXiv2003

On the irreducibility of multivariate subresultants

Let $P_1,...,P_n$ be generic homogeneous polynomials in $n$ variables of degrees $d_1,...,d_n$ respectively. We prove that if $ν$ is an integer satisfying ${\sum_{i=1}^n d_i}-n+1-\min\{d_i\}<ν,$ then all multivariate subresultants associated to the family $P_1,...,P_n$ in degree $ν$ are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of $\binom{ρ-ν+n-1}{n-1}$ smooth isolated points in $\PP^{n-1}.$

math.AG↗