Non-commutative Edmonds’ problem and matrix semi-invariants
Non-commutative Edmonds’ problem and matrix semi-invariants
复制标题
非交换埃德蒙兹问题和矩阵半不变量
DOI：
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发表时间：
2015
影响因子：
1.4
通讯作者：
K. Subrahmanyam
中科院分区：
文献类型：
--
作者：
G. Ivanyos;Youming Qiao;K. Subrahmanyam
AbstractIn 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an $${n \times n}$$n×n 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 Sémin 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 (Hrubeš 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 $${\mathbb{F}}$$F-algebra of matrix semi-invariants, denoted as R(n, m). For a field $${\mathbb{F}}$$F, it is the ring of invariant polynomials for the action of $${{\rm SL}(n, \mathbb{F}) \times
{\rm SL}(n, \mathbb{F})}$$SL(n,F)×SL(n,F) on tuples of matrices-$${(A, C)\in {\rm SL}(n, \mathbb{F}) \times
{\rm SL}(n, \mathbb{F})}$$(A,C)∈SL(n,F)×SL(n,F) sends $${(B_{1}, \ldots, B_m)\in M(n, \mathbb{F})^{\oplus m}}$$(B1,…,Bm)∈M(n,F)⊕m to $${(AB_1 {C}^{{\rm T}}, \ldots, AB_m {C}^{\rm T})}$$(AB1CT,…,ABmCT). 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 $${\leq \sigma}$$≤σ, then there follows a $${{\rm poly}(n,\sigma)}$$poly(n,σ)-time randomized algorithm to decide whether the non-commutative rank of T is full. To our knowledge, previously the best bound for $${\sigma}$$σ was $${O(n^{2}\cdot 4^{n^2})}$$O(n2·4n2) 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 $${\sigma}$$σ for R(n, m) over $${\mathbb{Q}}$$Q, deciding whether or not T has non-commutative rank < n over $${\mathbb{Q}}$$Q can be done deterministically in time polynomial in the input size and $${\sigma}$$σ.When $${\mathbb{F}}$$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 $${\sigma\leq (n+1)!}$$σ≤(n+1)!. 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 $${\sigma}$$σ for matrix semi-invariants over fields of positive characteristics. Furthermore, this does not require $${\mathbb{F}}$$F to be algebraically closed.
AbstractIn 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an $${n \times n}$$n×n 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 Sémin 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 (Hrubeš 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 $${\mathbb{F}}$$F-algebra of matrix semi-invariants, denoted as R(n, m). For a field $${\mathbb{F}}$$F, it is the ring of invariant polynomials for the action of $${{\rm SL}(n, \mathbb{F}) \times
{\rm SL}(n, \mathbb{F})}$$SL(n,F)×SL(n,F) on tuples of matrices—$${(A, C)\in {\rm SL}(n, \mathbb{F}) \times
{\rm SL}(n, \mathbb{F})}$$(A,C)∈SL(n,F)×SL(n,F) sends $${(B_{1}, \ldots, B_m)\in M(n, \mathbb{F})^{\oplus m}}$$(B1,…,Bm)∈M(n,F)⊕m to $${(AB_1 {C}^{{\rm T}}, \ldots, AB_m {C}^{\rm T})}$$(AB1CT,…,ABmCT). 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 $${\leq \sigma}$$≤σ, then there follows a $${{\rm poly}(n,\sigma)}$$poly(n,σ)-time randomized algorithm to decide whether the non-commutative rank of T is full. To our knowledge, previously the best bound for $${\sigma}$$σ was $${O(n^{2}\cdot 4^{n^2})}$$O(n2·4n2) 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 $${\sigma}$$σ for R(n, m) over $${\mathbb{Q}}$$Q, deciding whether or not T has non-commutative rank < n over $${\mathbb{Q}}$$Q can be done deterministically in time polynomial in the input size and $${\sigma}$$σ.When $${\mathbb{F}}$$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 $${\sigma\leq (n+1)!}$$σ≤(n+1)!. 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 $${\sigma}$$σ for matrix semi-invariants over fields of positive characteristics. Furthermore, this does not require $${\mathbb{F}}$$F to be algebraically closed.