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In this paper we relate the deformation method in invariant theory to spherical subgroups. Let be a reductive group, an affine -variety and a spherical subgroup. We show that whenever is affine and its semigroup of weights is saturated, the algebra of -invariant regular functions on has a -invariant filtration such that the associated graded algebra is the algebra of regular functions of some explicit horospherical subgroup of . The deformation method in its usual form, as developed...
A ring extension is said to be strongly affine if each -subalgebra of is a finite-type -algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if is a quasi-local ring of finite dimension, then is integrally closed and strongly affine if and only if is a Prüfer extension (i.e. is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown. Let be...
For a symmetric (= invariant under the action of a compact Lie group G) semialgebraic basic set C, described by s polynomial inequalities, we show, that C can also be written by s + 1 G-invariant polynomials. We also describe orbit spaces for the action of G by a number of inequalities only depending on the structure of G.
Known results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.
Let Δ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e-1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d-n, then we show that the ideal of Φn is also perfect, and we construct a covariant which...
L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu solved the stable equivalence problem for the polynomial ring k[x,y] when k is a field of characteristic 0. In this note we give an affirmative solution for an arbitrary field k.
Let
be a field of characteristic zero and G be a finite group of automorphisms of projective plane over
. Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field
is algebraically closed. In this paper we prove that
is rational for an arbitrary field
of characteristic zero.
We give an overview of recent results concerning kernels of triangular derivations of polynomial rings. In particular, we examine the question of finite generation in dimensions 4, 5, 6, and 7.
We study -actions of the form , where is the dual (to ) -variety. These actions are called the doubled ones. A geometric interpretation of the complexity of the action is given. It is shown that the doubled actions have a number of nice properties, if is spherical or of complexity one.
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