The maximal minors of a matrix indeterminates are universal Gröbner basis by theorem Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they not always Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders extensions to bases. These experiments motivated new implementation algorithm which organized in Singular script falls back on ...

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...

)( 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...

We present a new algorithm for computing a SAGBI basis up to an arbitrary degree for a subalgebra generated by a set of homogeneous polynomials. Our idea is based on linear algebra methods which cause a low level of complexity and computational cost. We then use it to solve the membership problem in subalgebras.

In this paper we study the relation between nonhomogeneous and homogeneous Sagbi bases. As a consequence, we present a general principle of computing Sagbi bases of a subalgebra and its homogenized subalgebra, which is based on passing over to homogenized generators.

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...

We introduce canonical bases for subalgebras of quotients of the commutative and non-commutative polynomial ring. A more complete exposition can be found in 4]. Canonical bases for subalgebras of the commutative polynomial ring were introduced by Kapur and Madlener (see 2]), and independently by Robbiano and Sweedler ((5]). Some notes on the non-commutative case can be found in 3]. Using the la...

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.