Polynomial degree bounds for matrix semi-invariants

Abstract

Abstract. We study the left-right action of SL n × SL n on m-tuples of n × n matrices with entries in an infinite field K. We show that invariants of degree n 2 − n define the null cone. Consequently, invariants of degree ≤ n 6 generate the ring of invariants if char(K) = 0. We also prove that for m ≫ 0, invariants of degree at least n⌊ √ n + 1⌋ are required to define the null cone. We generalize our results to matrix invariants of m-tuples of p × q matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for noncommutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials.1. Introduction 1.1. Degree bounds for invariant rings. Let Mat p,q be the set of p × q matrices with entries in an infinite field K. The group GL n acts on Mat m n,n by simultaneous conjugation. Procesi showed that in characteristic 0, the invariant ring is generated by traces of words in the matrices. Razmyslov ([35, final remark]) showed that that the invariant ring is generated by polynomials of degree ≤ n 2 by studying trace identities (see also [13]). In positive characteristic, generators of the invariant ring were given by Donkin in [14,15]. Domokos proved an upper bound O(n 7 m n ) for the degree of generators (see [10,11]). In this paper we will focus on the left-right action of G = SL n × SL n on the space V = Mat m n,n of m-tuples of n × n matrices. This action is given byThe group G also acts on the graded ring K[V ] of polynomial functions on V , and the subring of G-invariant polynomials is denoted by R(n, m)It is well-known that R(n, 1) is generated by the determinant det(X 1 ), and R(n, 2) is generated by the coefficients of det(X 1 + tX 2 ) as a polynomial in t. Because the group G is reductive, this invariant ring is finitely generated (see [20,21,30,19]). Definition 1.1. The number β(n, m) is the smallest nonnegative integer d such that R(n, m) is generated by invariants of degree ≤ d.The following bounds are known if K has characteristic 0:(1) β(n, 1) = β(n, 2) = n; (2) β(1, m) = 1;The first author was supported by NSF grant DMS-1302032 and the second author was supported by NSF grant DMS-1361789.The bounds in (1) follow from the descriptions of R(n, 1) and R(n, 2) above and (2) is trivial. The bound (3) can be found in [8] (see also [25]). This bound also follows from the First Fundamental Theorem of Invariant Theory for SO 4 , because SL 2 × SL 2 is a finite central extension of SO 4 and the representation Mat 2,2 of SL 2 × SL 2 corresponds to the standard 4-dimensional representation of SO 4 . The bound (4) was given in [9]. (5) and (6) were proved by the second author in [29]. Some explicit upper bounds for β(3, m) for m = 4, 5, 6, 7, 8 that are sharper than (5) were also given ...

Generating invariant rings of quivers in arbitrary characteristic

“…We showed the above results in characteristic 0 in [5,6]. In this paper, we show that the restrictions on characteristic can be removed.…”

“…While finding a minimal set of generators is perhaps too hard a question to answer, once could ask instead for a bound on the degree of generators. The methods of Popov and Derksen give us a general method to obtain bounds in characteristic 0, see [3,4,36,37]. Such a method does not exist in positive characteristic.…”

“…Knop showed that in certain cases, the degree is in fact atmost the negative of the Krull dimension of the invariant ring, and in some more special cases that the degree is equal to − dim V (see [29,30,4]). Further, there is a general method to obtain a set of invariants defining the null cone in characteristic 0, see [3]. Hence Theorem 2.3 can be used in characteristic 0 to give concrete bounds.…”

“…In the cases of interest to us, (1) and (3) will be a consequence of good filtrations, and (2) will be provided by the results in [5].…”

“…In [3], there is a general method for finding degree bounds in characteristic 0 (see also [4]). We analyze the role of characteristic 0 in the method and identify the necessary ingredients in order to adapt the method to positive characteristic.…”