arXiv:cs/0612050v2 [cs.SC] 15 Oct 2007 EXPLICIT FACTORS OF SOME ITERATED RESULTANTS AND DISCRIMINANTS LAURENT BUS ´ E AND BERNARD MOURRAIN dedicated to Professor Jean-Pierre Jouanolou Abstract. In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the in- put 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 discrim- inants 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 di- rect corollaries some results conjectured by Collins [9] and McCallum [26, 27] 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. 1. Introduction Resultants provide an essential tool in constructive algebra and in equation solv- ing, for projecting the solution of a polynomial system into a space of smaller di- mension. In the univariate case, a well-known construction due to J.J. Sylvester (1840) consists in eliminating the monomials 1,X,...,X m+n−1 in the multiples (X i P (X )) 0≤i≤n−1 , (X j Q(X )) 0≤j≤m−1 , of two given polynomials P,Q of degree m and n, and in taking the determinant of the corresponding (m+n) ×(m+n) matrix. Though the first resultant construction appeared probably in the work of E. B´ ezout [4] (see also Euler’s work), and although contemporary to related works (Jacobi 1835, Richelot 1840, Cauchy 1840, . . . ), this method remains well-known as Sylvester’s resultant. It is nowadays a fundamental tool used in effective algebra to eliminate a variable between two polynomials. It is a natural belief that equipped with such a tool which eliminates one vari- able at a time, one can iteratively eliminate several variables. This approach was actually exploited, for instance in [30], to deduce theoretical results (such as the ex- istence of eliminant polynomials) in several variables. However, if we are interested in structural results as well as practical computations or complexity issues, this Date : February 1, 2008. This work was first presented at the conference in honor of Jean-Pierre Jouanolou, held at Luminy, Marseille, 15-19 th May 2006. 1