Sparse Resultant of Composed Polynomials IMixed-. Unmixed Case

The main question of this paper is: What happens to sparse resultants under composition? More precisely, let f1, . . . , fn be homogeneous sparse polynomials in the variables y1, . . . , yn and g1, . . . , gn be homogeneous sparse polynomials in the variables x1, . . . , xn. Let fi ◦ (g1, . . . , gn) be the sparse homogeneous polynomial obtained from fi by 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, . . . , fn and g1, . . . , gn? The main contribution of this paper is to provide an answer for the case when g1, . . . , gn are unmixed, namely, ResC1,...,Cn (f1 ◦ (g1, . . . , gn) , . . . , fn ◦ (g1, . . . , gn)) = Resd1,...,dn (f1, . . . , fn) Vol(Q) ResB (g1, . . . , gn) d1···dn , where Res d1,...,dn stands for dense (Macaulay) resultant with respect to the total degrees di of the fi’s, ResB stands for unmixed sparse resultant Originally titled “A Chain Rule for Sparse Resultants”. Partially supported by NSF project “Computing with composed functions”, NSF CCR9972527