 
Summary: SAGBI AND SAGBIGR š
OBNER BASES OVER PRINCIPAL IDEAL DOMAINS
WILLIAM W. ADAMS, SERKAN HOSžTEN, PHILIPPE LOUSTAUNAU, J. LYN MILLER
1. Introduction
In 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 socalled
SAGBI (subalgebra analogues of Gršobner bases for ideals) bases for the subalgebras themselves and
SAGBIGršobner 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 SAGBIGršobner bases over any commutative
Noetherian ring we refer the reader to Miller [6].
In [6] algorithms are given for the computation of SAGBI and SG bases over an arbitrary Noe
therian commutative ring R. In addition to the usual Buchbergerstyle algorithms the algorithms
presented there relied on elimination order computations of Gršobner bases over R. When R is a
field, these extra Gršobner basis computations were replaced by computing the minimal Hilbert ba
sis for the set of solutions of certain linear diophantine equations. These in turn can be constructed
by Gršobner 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 Gršobner basis computations over R. Next
we go on to consider the same question for SG bases. Here we show that the elimination order
