On Polynomial Ideals, Their Complexity, and Applications

  title={On Polynomial Ideals, Their Complexity, and Applications},
  author={Ernst W. Mayr},
  • E. Mayr
  • Published in FCT 22 August 1995
  • Mathematics
A polynomial ideal membership problem is a (w+1)-tuple P=(f, g1,g2, ..., g w ) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known that this problem is exponential space complete. We discuss complexity results known for a number of problems related to polynomial ideals, like the word problem for commutative semigroups, a quantitative version… 
Exponential space computation of Gröbner bases
Using the ability to find normal forms, it is shown how to obtain the Groebner basis in exponential space by transforming a representation of the normal form into a system of linear equations and solving this system.
Encoding the dynamics of deterministic systems
We present a model for the dynamics of discrete deterministic systems, based on an extension of the Petri nets framework. Our model relies on the definition of a priority relation between conflicting


Membership in Plynomial Ideals over Q Is Exponential Space Complete
It is shown that the problem for multivariate polynomials over the rationals is solvable in exponential space, establishing its exponential space completeness.
The Structure of Polynomial Ideals and Gröbner Bases
It is shown that every ideal has a cone decomposition of a standard f orm and the following sharpened bound for the degree of polynomials in a Grobner basis can be produced.
Exponential space complete problems for Petri nets and commutative semigroups (Preliminary Report)
The uniform word problem for commutative semigroups (UWCS) is the problem of determining from any given finite set of defining relations and any pair of words, whether the words describe the same
Binomial Ideals
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many
The Complexity of the Boundedness, Coverability, and Selfcoverability Problems for Commutative Semigroups
In this paper, efficient decision procedures for the boundedness, coverability and selfcoverability problems for commutative semigroups are obtained. These procedures require at most space 2^{cn},
The complexity of the equivalence problem for commutative semigroups and symmetric vector addition systems
This paper shows that the equivalence problems for commutative semigroups and symmetric vector addition systems are decidable in space cNlogN for some fixed constant c, solving an open question by
Ideals, Varieties, and Algorithms
(here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In
Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.)
The algorithmic roots of algebraic object, called a close relationship between ideals, many of polynomial equations in geometric, object called a more than you, for teaching purposes and varieties, and the solutions and reduce even without copy.
A New Lower Bound Construction for Commutative Thue Systems with aApplications
  • C. Yap
  • Mathematics
    J. Symb. Comput.
  • 1991