Elimination theory in codimension one and applications

In these notes, we present a general framework to compute the codimension one part of the elimination ideal of a system of homogeneous polynomials. It is based on the computation of the so-called MacRae’s invariants that we will obtain by means of determinants of complexes. Our approach mostly uses tools from commutative algebra. We begin with some basics on elimination theory and then introduce the MacRae’s invariant and the so-called determinants of complexes. The rest of these notes illustrates our approach through two important examples: the Macaulay’s resultant of n homogeneous polynomials in n variables and the computation of an implicit equation of a parameterized hypersurface using syzygies. Key-words: Elimination theory, homogeneous polynomial systems, resultants, determinants of complexes, computational algebra, implicitization of rational hypersurfaces. Notes of lectures given at the CIMPA-UNESCO-IRAN school in Zanjan, Iran, July 9-22 2005. in ria -0 00 77 12 0, v er si on 3 4 M ar 2 01 3 Théorie de l’élimination en codimension un et applications Résumé : Dans ces notes, nous présentons une approche algébrique pour calculer la partie de codimension un d’un idéal d’élimination d’un système de polynômes homogènes. Elle repose principalement sur le calcul d’invariants dits de MacRae que nous obtiendrons en termes de déterminants de complexes. Dans un premier temps des résultats essentiels de la théorie de l’élimination sont rappelés puis les invariants de MacRae et les déterminants de complexes sont introduits. Le reste de ces notes illustre cette approche au travers de deux exemples: le résultant de Macaulay de n polynômes homogènes en n variables puis le calcul de l’équation implicite d’une hypersurface paramétrée en utilisant les syzygies de cette paramétrisation. Mots-clés : Théorie de l’élimination, systèmes de polynômes homogènes, résultants, déterminants de complexes, calcul formel, implicitation d’hypersurfaces rationnelles. in ria -0 00 77 12 0, v er si on 3 4 M ar 2 01 3 Elimination theory in codimension one and applications 3 In these notes, we present a general framework to compute the codimension one part of the elimination ideal of a system of homogeneous polynomials. It is based on the socalled MacRae’s invariants that can be obtained by means of determinants of complexes. Our approach mostly uses tools from commutative algebra and is inspired by the works of Jean-Pierre Jouanolou [22, 23, 25] (see also [31] for a similar point of view). We begin with some basics on elimination theory. Then, in section 2, we introduce the MacRae’s invariant and the so-called determinants of complexes that will allow us to compute this invariant. The rest of these notes illustrates our approach through two examples: the Macaulay’s resultant of n homogeneous polynomials in n variables and the computation of an implicit equation of a parameterized hypersurface using syzygies. The first one is treated in section 3 where we follow the monograph [22]. The second one, treated in section 4, report on joint works with Marc Chardin and Jean-Pierre Jouanolou [7, 4, 8]. All along the way, we will recall some tools from commutative algebra and algebraic geometry which may be useful for other purposes.