A Riemann hypothesis analogue for invariant rings

A Riemann hypothesis analogue for coding theory was introduced by I.M. Duursma [A Riemann hypothesis analogue for self-dual codes, in: A. Barg, S. Litsyn (Eds.), Codes and Association Schemes (Piscataway, NJ, 1999), American Mathematical Society, Providence, RI, 2001, pp. 115–124]. In this paper, we extend zeta polynomials for linear codes to ones for invariant rings, and we investigate whether a Riemann hypothesis analogue holds for some concrete invariant rings. Also we shall show that there is some subring of an invariant ring such that the subring is not an invariant ring but extremal polynomials all satisfy the Riemann hypothesis analogue.