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This paper contains three new algorithms for computing invariant rings. The first two apply to invariants of a finite group acting on a finitely generated algebra over a Euclidean ring. This may be viewed as a first step in “computational arithmetic invariant theory.” As a special case, the algorithms can compute multiplicative invariant rings. The third algorithm computes the invariant ring of a reductive group acting on a vector space, and often performs better than the algorithms known to date. The main tool upon which two of the algorithms are built is a generalized version of an ideal that was already used by Derksen in his algorithm for computing invariants of linearly reductive groups. As a further application, these so-called extended Derksen ideals give rise to invariantization maps, which turn an arbitrary ring element into an invariant. For the most part, the algorithms of this paper have been implemented. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group acting on a polynomial ring A over the field K and let denote the corresponding ring of invariants. Let B be the subalgebra of generated by all homogeneous elements of degree less than or equal to the group order . Then in general B is not equal to if the characteristic of K divides . However we prove that the field of fractions coincides with the field of invariants . We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for . We prove that there is always a nonzero transfer of degree , such that the localization can be generated by fractions of homogeneous invariants of degrees less than . If with finite-dimensional -module V, then c can be chosen in degree one and can be replaced by . Let denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that contains the transfer ideal and thus can be used to calculate generators for by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for . [Copyright &y& Elsevier]

Let A=BG, where B is a Noetherian algebra over a field K of characteristic p≠0 and G is a finite group such that p divides &z.sfnc;G&z.sfnc;. We give estimates for the depth of A in terms of the codimension of the branch locus of the extension B/A. [Copyright &y& Elsevier]