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Expressing a fraction of two determinants as a determinant

By Erich Kaltofen, Pascal Koiran, École Normale and Supérieure Lyon

Abstract

Suppose the polynomials f and g in K[x1,...,xr] over the field K are determinants of non-singular m×m and n×n matrices, respectively, whose entries are in K∪{x1,...,xr}. Furthermore, suppose h = f/g is a polynomial in K[x1,...,xr]. We construct an s × s matrix C whose entries are in K∪{x1,...,xr}, such that h = det(C) and s = γ(m+n) 6, where γ = O(1) if K is an infinite field or if for the finite field K = Fq with q elements we have m = O(q), and where γ = (log q m) 1+o(1) if q = o(m). Our construction utilizes the notion of skew circuits by Toda and weakly-skew circuits by Malod and Portier. Our problem was motivated by resultant formulas derived from Chow forms. Additionally, we show that divisions can be removed from formulas that compute polynomials in the input variables over a sufficiently large field within polynomial formula size growth

Publisher: ACM Press
Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.372.1503
Provided by: CiteSeerX
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