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JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
S 0894-0347(04)00451-5
Article electronically published on January 15, 2004
IDEAL MEMBERSHIP IN POLYNOMIAL RINGS
OVER THE INTEGERS
MATTHIAS ASCHENBRENNER
Introduction
The following well-known 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

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics