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It is well known that the ring of polynomial invariants of a reductive group is finitely generated. However, it is difficult to give strong upper bounds on the degrees of the generators, especially over fields of positive characteristic. In this paper, we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi-invariants in arbitrary characteristic, and consequently for matrix invariants. Our results generalize to invariants and semi-invariants of quivers. [ABSTRACT FROM AUTHOR]

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 non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials. [ABSTRACT FROM AUTHOR]

We study the ring R ( n , m ) of invariants for the left–right action of SL n × SL n on m -tuples of n × n complex matrices. We show that R ( 3 , m ) is generated by invariants of degree ≤309 for all m . Then, we use a combinatorial description of the invariants to show that R ( n , m ) cannot be generated by invariants of degree < n 2 for large m . We also compute the Hilbert series for several cases. [ABSTRACT FROM AUTHOR]