Non-commutative calculations are considered from the molecular computing point of view. The main idea is that one can get more advantage in using molecular computing for noncommutative computer algebra compared with a commutative one. The restrictions, connected with the coefficient handling in Gröbner basis calculations are investigated. Semigroup and group cases are considered as more appropr...

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.

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.

We present a characteristic-free algorithm for computing minimal generating sets of invariant rings of permutation groups. We circumvent the main weaknesses of the usual approaches (using classical Gröbner basis inside the full polynomial ring, or pure linear algebra inside the invariant ring) by relying on the theory of SAGBI-Gröbner basis. This theory takes, in this special case, a strongly c...

The theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial ordering which are not necessarily well ordering [1].) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Ana...

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...

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.