Sparse Resultant of Composed Polynomials I Mixed–Unmixed Case

✍ Scribed by Hoon Hong; Manfred Minimair

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 381 KB
• Volume: 33
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.2001.0516

✦ Synopsis

What happens to sparse resultants under composition? More precisely, let f 1 , . . . , fn be homogeneous sparse polynomials in the variables y 1 , . . . , yn and g 1 , . . . , gn be homogeneous sparse polynomials in the variables x 1 , . . . , xn. Let f i • (g 1 , . . . , gn) be the sparse homogeneous polynomial obtained from f i by replacing y j by g j . Naturally a question arises: Is the sparse resultant of f 1 • (g 1 , . . . , gn) , . . . , fn • (g 1 , . . . , gn) in any way related to the (sparse) resultants of f 1 , . . . , fn and g 1 , . . . , gn? The main contribution of this paper is to provide an answer for the case when g 1 , . . . , gn are unmixed, namely, Res C 1 ,...,Cn (f 1 • (g 1 , . . . , gn) , . . . , fn • (g 1 , . . . , gn))

where Res d 1 ,...,dn stands for the dense (Macaulay) resultant with respect to the total degrees d i of the f i 's, Res B stands for the unmixed sparse resultant with respect to the support B of the g j 's, Res C 1 ,...,Cn stands for the mixed sparse resultant with respect to the naturally induced supports C i of the f i •(g 1 , . . . , gn)'s, and Vol (Q) for the normalized volume of the Newton polytope of the g j . The above expression can be applied to compute sparse resultants of composed polynomials with improved efficiency.