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- W1973133099 abstract "Let $A(x) = (A_{ij} (x))$ be a square matrix with $A_{ij}$ being a polynomial in x. This paper proposes “combinatorial relaxation-” type algorithms for computing the degree of the determinant $delta (A) = deg_{x} det A(x)$ based on its combinatorial upper bound $widehat{delta}(A)$, which is defined in terms of the maximum weight of a perfect matching in an associated graph. The graph is bipartite for a general square matrix A and nonbipartite for a skew-symmetric A. The algorithm transforms A to another matrix $A^{prime}$, for which $delta (A) = delta (A^{prime}) = widehat{delta}(A^{prime})$ with successive elementary operations. The algorithm is efficient, making full use of the fast algorithms for weighted matchings; it is combinatorial in almost all cases (or generically) and invokes algebraic elimination routines only when accidental numerical cancellations occur. It is shown in passing that for a (skew-)symmetric polynomial matrix $A(x)$ there exists a unimodular matrix $U(x)$ such that $A'(x) = U(x)A(x)U(x)^{text{T}}$ satisfies $delta (A) = delta (A^{prime}) = widehat{delta}(A^{prime})$." @default.
- W1973133099 created "2016-06-24" @default.
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- W1973133099 date "1995-08-01" @default.
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- W1973133099 title "Computing the Degree of Determinants via Combinatorial Relaxation" @default.
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- W1973133099 doi "https://doi.org/10.1137/s0097539791201897" @default.
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