Denote by Rn;m the ring of invariants of m-tuples of n × n matrices (m; n ¿ 2) over an inÿnite base ÿeld 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 Rn;m. We extend this relationship to the case when the base ÿeld has positive characteristic. In particular, we show that if 0 ¡ char(K)) 6 n, then Rn;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 Rn;m that are complete intersections, known before when char(K) = 0, is valid for any inÿnite K. We give a Cohen-Macaulay presentation of R2;4, and analyze the di erence between the cases char(K) = 2 and char(K) = 2.