 
Summary: BOUNDS AND DEFINABILITY IN POLYNOMIAL RINGS
MATTHIAS ASCHENBRENNER
Abstract. We study questions around the existence of bounds and the de
pendence on parameters for linearalgebraic problems in polynomial rings over
rings of an arithmetic flavor. In particular, we show that the module of syzy
gies of polynomials f1, . . . , fn R[X1, . . . , XN ] with coefficients in a Pršufer
domain R can be generated by elements whose degrees are bounded by a num
ber only depending on N, n and the degree of the fj. This implies that if
R is a BŽezout domain, then the generators can be parametrized in terms of
the coefficients of f1, . . . , fn using the ring operations and a certain division
function, uniformly in R.
Introduction
The main theme of this article is the existence of bounds for basic operations of
linear algebra in polynomial rings over (commutative) rings of an arithmetic nature.
The following result, shown in Section 3 below, is typical.
Theorem A. Given integers N, d, n 0 there exists an integer = (N, d, n) with
the following property: for every Pršufer domain R and polynomials f1, . . . , fn
R[X] = R[X1, . . . , XN ] of (total) degree d, the R[X]submodule of R[X]n
con
sisting of all solutions to the linear homogeneous equation
