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 ...