Multivariate subresultants Original Research Article

In this text, we will introduce the natural generalization of the so-called subresultants of two polynomials in one variable, to the case of s ≤ n homogeneous polynomials in n variables. As a special case, we will of course recover the multivariate resultant. A first attempt in this direction was done by Gonzalez-Vega (1991). If P1, …, Ps are homogeneous polynomials of k[X1, …, Xn] with di = deg Pi > 0, and s ≤ n we define a polynomial ΔSv in the coefficients of the Piʹs attached to the following data: (i) the numbers n and s and the s-tuple D = (d1,…,ds), (ii) a positive integer v, and (iii) a set S of Hd(v) monomials of degree v, where Hd(v) is the Hubert function of a complete intersection given by s homogeneous polynomials in n variables of degrees d1, …, ds. The universal property of ΔSv is the following. If ψ is the canonical specialization homomorphism from the universal ring Z [coeff. of the Piʹs] to k sending each coefficient on its value, then if k is a field: ψ (ΔSv) ≠ 0 if and only if Iv + kleft angle bracketSright-pointing angle bracket = K[X1, …, Xn]v, where Iv is the degree v part of the ideal generated by the Piʹs. We recover the resultant, taking r = n, v > d1 + cdots, three dots, centered + dn − n and S = empty set︀ (Hd(v) = 0 for such a v). Moreover, we prove that for any monomial Xβ negated set membership S of degree v we have a “universal” relation image In the particular case r = N = 2, V = d1 + d2 − j − 1, Sj = {X2v, X2v − j + 1, X1, … X2v − j + 1 X1j − 1} (S0 = empty set︀) and Xβ = X2v − j X1j, the left side of this relation is the so-called subresultant of order j of p1 and P2. So, the classical subresultants are also a special case of these general objects.