 
Summary: JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
S 08940347(04)004515
Article electronically published on January 15, 2004
IDEAL MEMBERSHIP IN POLYNOMIAL RINGS
OVER THE INTEGERS
MATTHIAS ASCHENBRENNER
Introduction
The following wellknown theorem, due to Grete Hermann [20], 1926, gives an
upper bound on the complexity of the ideal membership problem for polynomial
rings over fields:
Theorem. Consider polynomials f0, . . . , fn F[X] = F[X1, . . . , XN ] of (total)
degree d over a field F. If f0 (f1, . . . , fn), then
f0 = g1f1 + · · · + gnfn
for certain g1, . . . , gn F[X] whose degrees are bounded by , where = (N, d)
depends only on N and d (and not on the field F or the particular polynomials
f0, . . . , fn).
This theorem was a first step in Hermann's project, initiated by work of Hentzelt
and Noether [19], to construct bounds for some of the central operations of com
mutative algebra in polynomial rings over fields. A simplified and corrected proof
