Let k be a field, Ln = k[x ±1 1 , . . . , x ±1 n ] be the Laurent polynomial ring in n variables and G be a group of k-algebra automorphisms of Ln. We give a necessary and sufficient condition for the ring of invariants Ln to have a SAGBI basis. We show that if this condition is satisfied then Ln has a SAGBI basis relative to any choice of coordinates in Ln and any term order.

In this paper we examine subalgebras on two generators in the univariate polynomial ring. A set, S, of polynomials in a subalgebra of a polynomial ring is called a canonical basis (also referred to as SAGBI basis) for the subalgebra if all lead monomials in the subalgebra are products of lead monomials of polynomials in S. In this paper we prove that a pair of polynomials {f, g} is a canonical ...

Let k be a field and G be a finite subgroup of GLn(Z). We show that the ring of multiplicative invariants k[x±1 1 , . . . , x ±1 n ] G has a finite SAGBI basis if and only if G is generated by reflections.

Our interest in the subject of this paper is inspired by Hong (1998), where Hoon Hong addresses the problem of the behavior of Gröbner bases under composition of polynomials. More precisely, let Θ be a set of polynomials, as many as the variables in our polynomial ring. The question then is under which conditions on these polynomials it is true that for an arbitrary Gröbner basis G (with respec...

Let Cp denote the cyclic group of order p where p ≥ 3 is prime. We denote by Vn the indecomposable n dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V2 ⊕ V2 ⊕ V3]p .

It is well-known, that the ring C[X1, . . . , Xn]n of polynomial invariants of the alternating group An has no finite SAGBI basis with respect to the lexicographical order for any number of variables n ≥ 3. This note proves the existence of a nonsingular matrix δn ∈ GL(n,C) such that the ring of polynomial invariants C[X1, . . . ,Xn] δn n , where An n denotes the conjugate of An with respect to...

)( was investigated. It turned out that only invariant rings of direct products of symmetric groups have a finite SAGBI basis, which is then, in addition, multilinear. Of course, it would be of interest to have such a strong characterization with respect to any other admissible order [4, 6]. To achieve this seems to be all but trivial. One step towards the understanding of the behavior of SAGBI...

In this paper, a new  algorithm for computing secondary invariants of  invariant rings of monomial groups is presented. The main idea is to compute simultaneously a truncated SAGBI-G basis and the standard invariants of the ideal generated by the set of primary invariants.  The advantage of the presented algorithm lies in the fact that it is well-suited to complexity analysis and very easy to i...