Some Complexity Results for Polynomial Ideals

@article{Mayr1997SomeCR,
  title={Some Complexity Results for Polynomial Ideals},
  author={Ernst W. Mayr},
  journal={J. Complex.},
  year={1997},
  volume={13},
  pages={303-325}
}
  • E. Mayr
  • Published 1 September 1997
  • Mathematics
  • J. Complex.
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)-tupleP= (f,g1,g2, ?gwwherefand thegiare multivariate polynomials, and the problem is to determine whetherfis in the ideal generated by thegi. For polynomials over the integers or rationals, this problem is known to be… 
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References

SHOWING 1-10 OF 93 REFERENCES
Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory
Problems connected with ideals generated by finite sets F of multivariate polynomials occur, as mathematical subproblems, in various branches of systems theory, see, for example, [5]. The method of
On the complexity of computing a Gröbner basis for the radical of a zero dimensional ideal
We show that if a system of polynomials f l , f 2 , . . . ,Jr in n variables with deg(fl) _< d over the rational numbers has only finitely many affine zeros, then, all the affine zeros can be
Membership in Plynomial Ideals over Q Is Exponential Space Complete
TLDR
It is shown that the problem for multivariate polynomials over the rationals is solvable in exponential space, establishing its exponential space completeness.
An optimal algorithm for constructing the reduced Gröbner basis of binomial ideals
TLDR
An optimal, exponential space algorithm for generating the reduced Grobner basis of binomial ideals over Q in general is presented, making use of the close relationship between commutative semigroups and pure difierence binomial ideal.
Lower Bounds for diophantine Approximation
TLDR
An intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero--dimensional polynomial equation system is obtained and represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals.
Membership problem, Representation problem and the Computation of the Radical for one-dimensional Ideals
TLDR
Borders are given for the complexity of the above problems which are simply exponential in the number n of variables in the one-dimensional case, and it is shown that in the general case the first two problems are doubly exponential only in the dimension of the ideal.
Exponential space computation of Gröbner bases
TLDR
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.
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
Optimal Gröbner Base Algorithms for Binomial Ideals
TLDR
An optimal, exponential space algorithm for generating the reduced Grobner basis of binomial ideals is exhibited and this result is then applied to derive space optimal decision procedures for the finite enumeration and subword problems for commutative semigroups.
Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.)
TLDR
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.
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