Rings of matrix invariants in positive characteristic

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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 n, m . 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 R 2, 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 R 2,4, and analyze the difference between the cases char( K)=2 and char( K)≠2.