Rings of matrix invariants in positive characteristics

Denote by R_n,_m the ring of invariants of m-tuples of n × n matrices (m,n ≥ 2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char(K) = 0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata-Higman theorem and the degree bound for generators of R_nm. We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0 < char(K)) ≤ n, then R_n,m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R₂,_m is determined for the case char(K)=2: it consists of 2^m+m-1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char(K) ≠2. We prove that the characterization of the R_n,_m that are complete intersections, known before when char(K) = 0, is valid for any infinite K. We give a Cohen-Macaulay presentation of R₂,₄, and analyze the difference between the cases char(K) = 2 and char(K) ≠2.