Explicit factors of some iterated resultants and discriminants

• DOI: 10.1090/s0025-5718-08-02111-x
• Publication Date: 2008-10-14T18:25:02Z
• Authors: Laurent Busé; Bernard Mourrain
• Keywords: [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC]; Computer Science - Symbolic Computation; FOS: Computer and information sciences; [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC]; [INFO.INFO-SC] Computer Science [cs]/Symbolic Computation [cs.SC]; [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]; Symbolic Computation (cs.SC); Mathematics - Commutative Algebra; Commutative Algebra (math.AC); 01 natural sciences; [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC]; Mathematics - Algebraic Geometry; FOS: Mathematics; [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]; 0101 mathematics; Algebraic Geometry (math.AG)

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.<br/>Selected for presentation at the conference MEGA 2007 (Strobl, Austria, June 25th - 29th)<br/>