ISSN:
1432-0622
Keywords:
Ascending chain condition
;
E-bases
;
Gröbner bases
;
Ideal membership problem
;
Rapidly growing functions
;
Ring of polynomials over the integers
;
S-polynomials
;
syzygies
;
Wainer hierarchy
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
,
Technology
Notes:
Abstract Consider ℤ[x 1,...,x n], the multivariate polynomial ring over integers involvingn variables. For a fixedn, we show that the ideal membership problem as well as the associated representation problem for ℤ[x 1,...,x n] are primitive recursive. The precise complexity bounds are easily expressible by functions in the Wainer hierarchy. Thus, we solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers. As a direct consequence, we also obtain a solution to certain foundational problem intrinsic to Kronecker's programme for constructive mathematics and provide an effective version of Hilbert's basis theorem. Our original interest in this area was aroused by Edwards' historical account of theKronecker's problem in the context of Kronecker's version of constructive mathematics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01188747
Permalink
