Efficient algorithms for ideal operations (extended abstract)

Journal of Symbolic Computation, 2000

This paper presents an algorithm for the Quillen-Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k x 0 ; :::;xn]=I, with I a monomial ideal and k a eld. T. Vorst proved that nitely generated projective modules over such algebras are free. Given a nitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis for P.

Central European Journal of Mathematics, 2011

We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.

Journal of Complexity, 1997

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g 1 , g 2 , . . . , g w ) where f and the g i are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases. © 1997 Academic Press

Groups, complexity, cryptology, 2024

For a finite Z-algebra R, i.e., for a Z-algebra which is a finitely generated Z-module, we assume that R is explicitly given by a system of Z-module generators G, its relation module Syz(G), and the structure constants of the multiplication in R. In this setting we develop and analyze efficient algorithms for computing essential information about R. First we provide polynomial time algorithms for solving linear systems of equations over R and for basic ideal-theoretic operations in R. Then we develop ZPP (zero-error probabilitic polynomial time) algorithms to compute the nilradical and the maximal ideals of 0-dimensional affine algebras K[x1,. .. , xn]/I with K = Q or K = Fp. The task of finding the associated primes of a finite Z-algebra R is reduced to these cases and solved in ZPPIF (ZPP plus one integer factorization). With the same complexity, we calculate the connected components of the set of minimal associated primes minPrimes(R) and then the primitive idempotents of R. Finally, we prove that knowing an explicit representation of R is polynomial time equivalent to knowing a strong Gröbner basis of an ideal I such that R = Z[x1,. .. , xn]/I.

Computational Commutative and Non- …, 2005

Abstract. Based on an explicit description of the idealization of a graded submodule of a graded free module, we examine the behaviour of Gröbner bases and minimal homogeneous systems of generators under this process. Then we show how one can ...

Lecture Notes in Computer Science, 1997

it The authors acknowledge the support of ESPRIT reactive LTR project 21024 FRISCO Lemma 1. Let I be an ideal of a ring R, a ∈ R, and assume that I : * (a) = I : (a n); then I = (I : (a n)) ∩ (I + (a n)).

Journal of Algebra, 1983

Computing Research Repository, 2011

We demonstrate a method to parallelize the computation of a Gr\"obner basis for a homogenous ideal in a multigraded polynomial ring. Our method uses anti-chains in the lattice $\mathbb N^k$ to separate mutually independent S-polynomials for reduction.

Journal of Symbolic Computation, 2019

In this paper, we present an algorithm for computing the minimal reductions of m-primary ideals of Cohen-Macaulay local rings. Using this algorithm, we are able to compute the Hilbert-Samuel multiplicities and solve the membership problem for the integral closure of m-primary ideals.

Gröbner Bases, Coding, and Cryptography, 2009