Polynomial bounds for rings of invariants

Hilbert proved that invariant rings are finitely generated for linearly red uctive groups acting rationally on a finite dimensional vector space. POPOV gave an explicit upper bound for the smallest integer d such that the inva riants of degree less than or equal to d generate the invariant ring. This bound has factorial growth. In this paper we will give a bound which depend s only polynomially on the input data.