Fast and deterministic computation of the determinant of a polynomial matrix

Given a square, nonsingular matrix of univariate polynomials $\mathbf{F}\in\mathbb{K}[x]^{n\times n}$ over a field $\mathbb{K}$, we give a deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm is $\bigO \left(n^{\omega}s\right)$ field operations where $s$ is the average column degree or the average row degree of $\mathbf{F}$. Here $\bigO$ notation is Big-$O$ with log factors omitted and $\omega$ is the exponent of matrix multiplication.

[1]  B. Beckermann,et al.  A Uniform Approach for the Fast Computation of Matrix-Type Padé Approximants , 1994, SIAM J. Matrix Anal. Appl..

[2]  Patrick Davies Howard Cheng.  Computing Popov Form of Ore Polynomial Matrices , 2006 .

[3]  George Labahn,et al.  A deterministic algorithm for inverting a polynomial matrix , 2015, J. Complex..

[4]  Wei Zhou,et al.  Fast Order Basis and Kernel Basis Computation and Related Problems , 2013 .

[5]  George Labahn,et al.  Computing minimal nullspace bases , 2012, ISSAC.

[6]  Arne Storjohann,et al.  On lattice reduction for polynomial matrices , 2000 .

[7]  Claude-Pierre Jeannerod,et al.  On the complexity of polynomial matrix computations , 2003, ISSAC '03.

[8]  Arne Storjohann,et al.  High-order lifting and integrality certification , 2003, J. Symb. Comput..

[9]  Erich Kaltofen,et al.  On computing determinants of matrices without divisions , 1992, ISSAC '92.

[10]  Mark Giesbrecht,et al.  Computing the Hermite Form of a Matrix of Ore Polynomials , 2011, ArXiv.

[11]  A. Storjohann.  Algorithms for matrix canonical forms , 2000 .

[12]  George Labahn,et al.  Efficient algorithms for order basis computation , 2012, J. Symb. Comput..

[13]  George Labahn,et al.  Unimodular completion of polynomial matrices , 2014, ISSAC.

[14]  George Labahn,et al.  Shifted normal forms of polynomial matrices , 1999, ISSAC '99.

[15]  George Labahn,et al.  Normal forms for general polynomial matrices , 2006, J. Symb. Comput..

[16]  Erich Kaltofen,et al.  ON THE COMPLEXITY OF COMPUTING DETERMINANTS , 2001 .

[17]  J. Hopcroft,et al.  Triangular Factorization and Inversion by Fast Matrix Multiplication , 1974 .

[18]  George Labahn,et al.  Computing column bases of polynomial matrices , 2013, ISSAC '13.

[19]  Manuel Bronstein,et al.  Fast deterministic computation of determinants of dense matrices , 1999, ISSAC '99.

[20]  Gilles Villard,et al.  On computing the determinant and Smith form of an integer matrix , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.