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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]

The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's [7] constructive characterization of the Krull dimension. While Lombardi's characterization uses the lexicographic monomial order, ours uses the graded (reverse) lexicographic order or, in fact, any graded rational monomial order. Apart from this, the paper contains some related results and some examples which readers may find illuminating. [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: We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can show that is the maximum number of analytically independent elements in an arbitrary ideal J of a local ring R and that if are (not necessarily finitely generated) subalgebras of a finitely generated algebra over a Noetherian Jacobson ring. [Copyright &y& Elsevier]

Abstract: Given a linear action of a group G on a K-vector space V, we consider the invariant ring , where is the dual space. We are particularly interested in the case where and G is the group of all upper unipotent matrices or the group of all upper triangular matrices in . In fact, we determine for and . The result is a complete intersection for all values of n and q. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of “doubly parametrized” series of group actions whose invariant rings are known to have a uniform description. [Copyright &y& Elsevier]

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]

Abstract: Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes in terms of a so-called colon-operation. From this, generators of can be obtained in finite time if it is finitely generated. Under the additional hypothesis that is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is . Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal. [Copyright &y& Elsevier]

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]

One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations.In the second part of the paper, the distribution of areas of sub-triangles is used for characterizing point configurations. Again it turns out that most configurations are reconstructible from the distribution of areas, though there are counterexamples. [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]

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]