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Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f ∈ K [ X ] = : R and computes the invariant ring (R f) G together with a presentation. The morphism X f → Spec ((R f) G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial ring, the algorithm requires no Gröbner basis computations. If R is a complete intersection, then so is (R f) G. [ABSTRACT FROM AUTHOR]

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: This paper studies separating subsets of an invariant ring or, more generally, of any set consisting of functions. We prove that a subset of a finitely generated algebra always contains a finite separating subset. We also show that a general version of Noether’s degree bound holds for separating invariants, independently of the characteristic. While the general finiteness result is non-constructive, the Noether bound provides an easy algorithm for computing separating invariants of finite groups. The paper also contains a conceptual investigation of the difference between separating and generating subsets. [Copyright &y& Elsevier]

We propose an algorithm for computing the radical of a polynomial ideal in positive characteristic. The algorithm does not involve polynomial factorization. [Copyright &y& Elsevier]