Sparse Resultant of Composed Polynomials I Mixed–Unmixed Case

AbstractThe main question of this paper is: What happens to sparse resultants under composition? More precisely, let f1,⋯, fnbe homogeneous sparse polynomials in the variables y1,⋯, ynand g1,⋯, gnbe homogeneous sparse polynomials in the variables x1,⋯, xn. Let fi ∘ (g1,⋯, gn) be the sparse homogeneous polynomial obtained from fiby replacing yj by gj. Naturally a question arises: Is the sparse resultant of f1 ∘ (g1,⋯, gn), ⋯, fn ∘ (g1,⋯, gn)in any way related to the (sparse) resultants of f1,⋯, fnand g1,⋯, gn? The main contribution of this paper is to provide an answer for the case when g1,⋯, gnare unmixed, namely, ResC1,⋯, Cn(f1 ∘ (g1,⋯, gn),⋯, fn ∘ (g1,⋯, gn)) = Resd1,⋯, dn(f1,⋯, fn)Vol(Q)ResB(g1,⋯, gn)d1⋯dn, where Resd1,..., dnstands for the dense (Macaulay) resultant with respect to the total degrees diof the fi’s, ResBstands for the unmixed sparse resultant with respect to the support B of the gj’s, ResC1,..., Cnstands for the mixed sparse resultant with respect to the naturally induced supports Ciof the fi ∘ (g1,⋯, gn)’s, and Vol(Q)for the normalized volume of the Newton polytope of the gj. The above expression can be applied to compute sparse resultants of composed polynomials with improved efficiency