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• W3081136268 abstract "Abstract In this paper, we address computation of the degree $$deg {rm Det} A$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>deg</mml:mo> <mml:mi>Det</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> of Dieudonné determinant $${rm Det} A$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Det</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> of $$begin{aligned} A = sum_{k=1}^m A_k x_k t^{c_k}, end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>m</mml:mi> </mml:munderover> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msup> <mml:mi>t</mml:mi> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $$A_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> are $$n times n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> matrices over a field $$mathbb{K}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>K</mml:mi> </mml:math> , $$x_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> are noncommutative variables, t is a variable commuting with $$x_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> , $$c_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> are integers, and the degree is considered for t . This problem generalizes noncommutative Edmonds' problem and fundamental combinatorial optimization problems including the weighted linear matroid intersection problem. It was shown that $$deg {rm Det} A$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>deg</mml:mo> <mml:mi>Det</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> is obtained by a discrete convex optimization on a Euclidean building (Hirai 2019). We extend this framework by incorporating a cost-scaling technique and show that $$deg {rm Det} A$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>deg</mml:mo> <mml:mi>Det</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> can be computed in time polynomial of $$n,m,log_2 C$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mo>log</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> , where $$C:= max_k |c_k|$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>max</mml:mo> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We give a polyhedral interpretation of $$deg {rm Det}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>deg</mml:mo> <mml:mi>Det</mml:mi> </mml:mrow> </mml:math> , which says that $$deg {rm Det}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>deg</mml:mo> <mml:mi>Det</mml:mi> </mml:mrow> </mml:math> A is given by linear optimization over an integral polytope with respect to objective vector $$c = (c_k)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Based on it, we show that our algorithm becomes a strongly polynomial one. We also apply our result to an algebraic combinatorial optimization problem arising from a symbolic matrix having $$2 times 2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> -submatrix structure." @default.
• W3081136268 created "2020-09-01" @default.
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• W3081136268 date "2022-07-23" @default.
• W3081136268 modified "2023-10-14" @default.
• W3081136268 title "A cost-scaling algorithm for computing the degree of determinants" @default.
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