Constructive noncommutative rank computation in deterministic polynomial time over fields of arbitrary characteristics

We extend our techniques developed in our manuscript mentioned in the subtitle to obtain a deterministic polynomial time algorithm for computing the non-commutative rank together with certificates of linear spaces of matrices over sufficiently large base fields. The key new idea is a reduction procedure that keeps the blow-up parameter small, and there are two methods to implement this idea: the first one is a greedy argument that removes certain rows and columns, and the second one is an efficient algorithmic version of a result of Derksen and Makam. Both methods rely crucially on the regularity lemma in the aforementioned manuscript, and in this note we improve that lemma by removing a coprime condition there. arXiv:1508.00690

[1]  Youming Qiao,et al.  Non-commutative Edmonds’ problem and matrix semi-invariants , 2015, computational complexity.

[2]  Harm Derksen,et al.  Polynomial bounds for rings of invariants , 2000 .

[3]  Joe W. Harris,et al.  Vector spaces of matrices of low rank , 1988 .

[4]  S. A. Amitsur.  Rational identities and applications to algebra and geometry , 1966 .

[5]  Tata Institute of Fundamental Research, Bombay, India , 1971 .

[6]  Avi Wigderson,et al.  A Deterministic Polynomial Time Algorithm for Non-commutative Rational Identity Testing , 2015, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[7]  Tsit Yuen Lam,et al.  A first course in noncommutative rings , 2002 .

[8]  Jstor,et al.  Proceedings of the American Mathematical Society , 1950 .

[9]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[10]  K. Wong.  The eigenvalue problem λTx + Sx , 1974 .

[11]  Christophe Reutenauer,et al.  COMMUTATIVE/NONCOMMUTATIVE RANK OF LINEAR MATRICES AND SUBSPACES OF MATRICES OF LOW RANK , 2004 .

[12]  A. Schofield,et al.  SKEW FIELDS: THEORY OF GENERAL DIVISION RINGS (Encyclopedia of Mathematics and its Applications 57) , 1998 .

[13]  Harm Derksen,et al.  Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients , 2000 .

[14]  Jan Draisma,et al.  The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms , 2006 .

[15]  Harm Derksen,et al.  Polynomial degree bounds for matrix semi-invariants , 2015, ArXiv.

[16]  Marek Karpinski,et al.  Deterministic Polynomial Time Algorithms for Matrix Completion Problems , 2010, SIAM J. Comput..

[17]  J. Edmonds.  Systems of distinct representatives and linear algebra , 1967 .

[18]  Michel Van den Bergh,et al.  Semi-invariants of quivers for arbitrary dimension vectors , 1999 .

[19]  M. D. ATKINSON,et al.  Primitive spaces of matrices of bounded rank. II , 1981, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[20]  Leonid Gurvits,et al.  Classical complexity and quantum entanglement , 2004, J. Comput. Syst. Sci..

[21]  Mátyás Domokos,et al.  Semi-invariants of quivers as determinants , 2001 .

[22]  P. Cohn.  Free rings and their relations , 1973 .

[23]  Jeffrey Shallit,et al.  The Computational Complexity of Some Problems of Linear Algebra , 1996, J. Comput. Syst. Sci..