The infiniteness of the SAGBI bases for certain invariant rings.
The concept of initial ideals for ideals of a polynomial ring in Gröbner bases theory is generalized in a natural way for subalgebras of a polynomial ring, and they are called initial algebras. A set of generators of a subalgebra is called a SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis if their initial monomials generate the initial algebra. The main difference between the initial ideal and the initial algebra is that the former always is finitely generated while the latter does not. Hence it is an important problem to find a criterion for the finite generation of initial algebras. \textit{M. Göbel} [J. Symb. Comput. 26, 261--272 (1998; Zbl 0916.13011)], studied this problem for the subalgebra of invariants of a permutation group \(G\). He proved that, with respect to the lexicographic order, the initial algebra of \(K[V]^G\) is finitely generated if and only if \(G\) is a direct product of symmetric groups. In the present paper, the same result is proved for any multiplicative order, i.e., a monomial order which does not require the minimality of the unit \(1\), and also for the action on the Laurent polynomial ring. A principal role in the proof plays the topological structure on the set of multiplicative orders. Using this structure and the Baire theorem, the author proves that there exist uncountable cardinality of distinct initial algebras for each invariant ring, when \(G\) is not a direct product of symmetric groups. (Note that in the case of initial ideals, there exist only a finite cardinality of distinct initial ideals for an ideal under a certain condition, although there exist infinitely many orders.) If \(G\) is a product of symmetric groups, the number of distinct initial algebras of the subalgebra of invariants is finite and is equal to the order of \(G\).
subalgebra analogue to Gröbner bases for ideals
subalgebra of invariants of a permutation group
scientific article; zbMATH DE number 1836041