Expressing a Fraction of Two Determinants as a Determinant.

Abstract : Suppose the multivariate polynomials f and g are determinants of non-singular matrices A and B whose entries are variables or field elements. Furthermore, suppose that the quotient h = f/g is also a polyonmial. We construct a matrix C such that h=det(C). The entries of C are variables or field elements, and its size is polynomial in size(A)+size(B). Our construction utilizes the notion of skew circuits by Toda and weakly 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.
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Contributor : Pascal Koiran <>
Submitted on : Friday, February 1, 2008 - 1:02:28 PM
Last modification on : Wednesday, August 21, 2019 - 10:30:06 AM
Long-term archiving on : Thursday, September 27, 2012 - 5:47:57 PM

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Pascal Koiran, Erich Kaltofen. Expressing a Fraction of Two Determinants as a Determinant.. 2008. ⟨ensl-00232169⟩

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