A finite universal SAGBI basis for the kernel of a derivation

Abstract

To find and to calculate generating sets for invariant rings i s a fundamental prob- lem in invariant theory with a long tradition. With the progress of computers, the sig- nificance of computational methods in this field has increase d. The SAGBI bases are the sets of generators of a subalgebra of a polynomial ring which have certain com- putational property. These are the natural Subalgebra Analogue to Grobner Bases for Ideals introduced at the end of 1980's by Robbiano and Sweed ler (20) and Kapur and Madlener (8), independently. There are indeed some applications of the SAGBI bases to invariant theory. The algorithm of Stillman and Tsai (23) gives a method for computing generating sets for certain invariant rings by using this notion. However, compared with the theory of Grobner bases, that of SAGBI bases has made a slow progress, and many basic problems remaining unsolved. The purpose of this paper is to investigate the properties of a SAGBI basis for the kernel of a derivation on a poly- nomial ring. The kernel of a derivation on a polynomial ring is closely related to an invari- ant ring. It is an important object in the study of invariant t heory and the fourteenth problem of Hilbert. It is well-known that some kind of derivation corresponds to an action of one-dimensional additive group, and the kernel and the invariant subring are the same. Moreover, various counterexamples to the fourteenth problem of Hilbert can be described as the kernel of a derivation. Nagata's counter example (17) and Roberts' counterexample (22) were described as this by Derksen (2) and by Deveney and Fin- ston (4), respectively. Nowicki showed that the invariant subring for a linear action of a connected linear algebraic group on a polynomial ring is obtained as the kernel of a derivation (18). Recently, new counterexamples to the fourteenth problem of Hilbert were constructed by using the kernel of a derivation by several people (cf. (1), (6), (10), (13)). We believe that a computational methods will give us further progress in this field. In this paper, is always a field of characteristic zero except Section 6. Let