Sparse Resultant of Composed Polynomials I Mixed–Unmixed Case

The 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, , ynandg1, , 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, , fnandg1, , 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.