arXiv subjects

Jean-Pierre Jouanolou

Publications and source records attributed to Jean-Pierre Jouanolou.

5 recordsLinked to original sources

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↗

arXiv2012

The eventual stability of depth, associated primes and cohomology of a graded module

The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of the degree from which it stabilizes. Before we establish these regularity results, we prove several facts about depth and cohomological dimension with respect to a finitely generated ideal and about Castelnuovo-Mumford regularity of a graded module.

math.AC↗

arXiv2007

Torsion of the symmetric algebra and implicitization

Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of the MacRae's invariant of a graded part of the symmetric algebra is optimal. Then we show that the extraneous factor that may appear in the process splits into a product a linear forms in the algebraic closure of the base field, each linear form being associated to a non complete intersection base point. Finally, we make a link between this method and a resultant computation for the case of rational plane curves and space surfaces.

math.AC↗

arXiv2006

An explicit duality for quasi-homogeneous ideals

Given r>=n quasi-homogeneous polynomials in n variables, the existence of a certain duality is shown and explicited in terms of generalized Morley forms. This result, that can be seen as a generalization of [3,corollary 3.6.1.4] (where this duality is proved in the case r=n), was observed by the author at the same time. We will actually closely follow the proof of (loc. cit.) in this paper.

math.AC↗

arXiv2003

On the closed image of a rational map and the implicitization problem

In this paper, we investigate some topics around the closed image $S$ of a rational map $λ$ given by some homogeneous elements $f_1,...,f_n$ of the same degree in a graded algebra $A$. We first compute the degree of this closed image in case $λ$ is generically finite and $f_1,...,f_n$ define isolated base points in $\Proj(A)$. We then relate the definition ideal of $S$ to the symmetric and the Rees algebras of the ideal $I=(f_1,...,f_n) \subset A$, and prove some new acyclicity criteria for the associated approximation complexes. Finally, we use these results to obtain the implicit equation of $S$ in case $S$ is a hypersurface, $\Proj(A)=\PP^{n-2}_k$ with $k$ a field, and base points are either absent or local complete intersection isolated points.

math.AG↗