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- W2071679970 abstract "The main question of this paper is: What happens to sparse resultants under composition? More precisely, let f 1 ,⋯, f n be homogeneous sparse polynomials in the variables y 1 ,⋯, y n and g 1 ,⋯, g n be homogeneous sparse polynomials in the variables x 1 ,⋯, x n . Let f i ∘ ( g 1 ,⋯, g n ) 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 ,⋯, g n ), ⋯, f n ∘ ( g 1 ,⋯, g n )in any way related to the (sparse) resultants of f 1 ,⋯, f n and g 1 ,⋯, g n ? The main contribution of this paper is to provide an answer for the case when g 1 ,⋯, g n are unmixed, namely, Res C 1 ,⋯, C n ( f 1 ∘ ( g 1 ,⋯, g n ),⋯, f n ∘ ( g 1 ,⋯, g n )) = Res d 1 ,⋯, d n ( f 1 ,⋯, f n ) Vol( Q ) Res B ( g 1 ,⋯, g n ) d 1 ⋯ d n , where Res d 1 ,..., d n 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 ,..., C n stands for the mixed sparse resultant with respect to the naturally induced supports C i of the f i ∘ ( g 1 ,⋯, g n )’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." @default.
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- W2071679970 date "2002-04-01" @default.
- W2071679970 modified "2023-09-30" @default.
- W2071679970 title "Sparse Resultant of Composed Polynomials I Mixed–Unmixed Case" @default.
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- W2071679970 doi "https://doi.org/10.1006/jsco.2001.0516" @default.
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