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Explicit factors of some iterated resultants and discriminants

Authors: Laurent Busé and Bernard Mourrain
Journal: Math. Comp. 78 (2009), 345-386
MSC (2000): Primary 13P05
Published electronically: April 16, 2008
MathSciNet review: 2448711
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Abstract: 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 (1975) and McCallum (1999, 2001 preprint) 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.

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Additional Information

Laurent Busé
Affiliation: GALAAD, INRIA, B.P. 93, 06902 Sophia Antipolis, France

Bernard Mourrain
Affiliation: GALAAD, INRIA, B.P. 93, 06902 Sophia Antipolis, France

Received by editor(s): December 19, 2006
Received by editor(s) in revised form: November 10, 2007
Published electronically: April 16, 2008
Additional Notes: This work was first presented at the conference in honor of Jean-Pierre Jouanolou, held at Luminy, Marseille, May 15–19, 2006
Dedicated: Dedicated to Professor Jean-Pierre Jouanolou
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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