Sagbi And Sagbi-Gröbner Bases Over Principal Ideal Domains

this paper we will discuss computational and structural properties of subalgebras of polynomial rings when the base ring is a principal ideal domain (PID). The objects we study are the so-called SAGBI (subalgebra analogues of Grobner bases for ideals) bases for the subalgebras themselves and SAGBI-Grobner bases for the ideals in the subalgebras (SG bases). We will discuss how to compute these objects, and our goal is to avoid computations over the PID as much as possible. Further we will show the existence of strong SAGBI bases for these subalgebras and give an algorithm to compute them. For the general theory of SAGBI and SAGBI-Grobner bases over any commutative Noetherian ring we refer the reader to Miller [7]. In [7] algorithms are given for the computation of SAGBI and SG bases over an arbitrary Noetherian commutative ring R. In addition to the usual Buchberger-style algorithms the algorithms presented there relied on elimination order computations of Grobner bases over R. When R is a field, these extra Grobner basis computations were replaced by computing the minimal Hilbert basis for the set of solutions of certain linear diophantine equations. These in turn can be constructed by Grobner basis techniques, but over a field. In this paper first we show that in the construction of SAGBI bases over R, a PID, we can avoid these extra Grobner basis computations over R. Next we go on to consider the same question for SG bases. Here we show that the elimination order computations over R can be replaced by a degree reverse lexicographic (degrevlex) computation over the same ring R. In the last section we will show that strong SAGBI bases, the analogue of strong Grobner bases for ideals in polynomial rings, always exist, and we will give an algorithm for their construction...