Non-commutative Edmonds' problem and matrix semi-invariants

作者：Ivanyos Gabor^*; Qiao Youming; Subrahmanyam K V

来源：Computational Complexity, 2017, 26(3): 717-763.

DOI：10.1007/s00037-016-0143-x

• Edmonds' problem
• symbolic determinant identity test
• semi-invariants of quivers
• non-commutative rank

In 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an matrix T with integral homogeneous linear polynomials. In this paper, we consider the non-commutative version of Edmonds' problem: compute the rank of T over the free skew field. This problem has been proposed, sometimes in disguise, from several different perspectives in the study of, for example, the free skew field itself (Cohn in J Symbol Log 38(2):309-314, 1973), matrix spaces of low rank (Fortin-Reutenauer in S,min Lothar Comb 52:B52f 2004), Edmonds' original problem (Gurvits in J Comput Syst Sci 69(3):448-484, 2004), and more recently, non-commutative arithmetic circuits with divisions (Hrube and Wigderson in Theory Comput 11:357-393, 2015. doi: 10.4086/toc.2015.v011a014)
It is known that this problem relates to the following invariant ring, which we call the -algebra of matrix semi-invariants, denoted as R(n, m). For a field , it is the ring of invariant polynomials for the action of on tuples of matrices- sends to . Then those T with non-commutative rank < n correspond to those points in the nullcone of R(n, m). In particular, if the nullcone of R(n, m) is defined by elements of degree , then there follows a -time randomized algorithm to decide whether the non-commutative rank of T is full. To our knowledge, previously the best bound for was over algebraically closed fields of characteristic 0 (Derksen in Proc Am Math Soc 129(4):955-964, 2001).
We now state the main contributions of this paper:
We observe that by using an algorithm of Gurvits, and assuming the above bound for R(n, m) over , deciding whether or not T has non-commutative rank < n over can be done deterministically in time polynomial in the input size and sigma.
When F is large enough, we devise an algorithm for the non-commutative Edmonds problem which runs in time polynomial in (n + 1)!. Furthermore, due to the structure of this algorithm, we also have the following results.
If the commutative rank and the non-commutative rank of T differ by a constant there exists a randomized efficient algorithm to compute the non-commutative rank of T. This improves upon a result of Fortin and Reutenauer, who gave a randomized efficient algorithm to decide whether the commutative and non-commutative ranks are equal.
We show that . This not only improves the bound obtained from Derksen's work over algebraically closed field of characteristic 0 but, more importantly, also provides for the first time an explicit bound on for matrix semi-invariants over fields of positive characteristics. Furthermore, this does not require to be algebraically closed.

• 出版日期2017-9