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} }
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…
71 Citations
Complexity of Membership Problems of Different Types of Polynomial Ideals
- Mathematics
- 2017
We survey degree bounds and complexity classes of the word problem for polynomial ideals and related problems. The word problem for general polynomial ideals is known to be exponential…
Radicals of Binomial Ideals and Commutative Thue Systems
- Mathematics, Computer Science
- 2017
This work presents a new algorithm for computing the radical of a binomial ideal which uses binomials as intermediate results of the computations only, but matches the running time of the best known algorithms.
The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties
- Mathematics, Computer ScienceFound. Comput. Math.
- 2007
It is shown that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced inPolynomial time to the problemof counting the number of complex common zeros of a finite set of multivariate polynomials.
On the complexity of counting components of algebraic varieties
- Mathematics, Computer ScienceJ. Symb. Comput.
- 2009
On the Complexity of Counting Irreducible Components and Computing Betti Numbers of Algebraic Varieties
- Mathematics, Computer Science
- 2007
This thesis gives a uniform method for the two problems #CCC and #ICC of counting the connected and irreducible components of complex algebraic varieties and proves that the problem of deciding connectedness of a complex affine or projective variety given over the rationals is PSPACE-hard.
The complexity of computing the Hilbert polynomial of smooth complex projective varieties
- Mathematics, Computer Science
- 2004
The main goal of this paper is to prove a stronger result: it is shown that the problem HILBERTsm of computing the Hilbert polynomial of a smooth complex projective variety V P n can be reduced inPolynomial time to the problem #HNC of counting the number of complex common zeros of a given finite set of complex polynomials.
Gröbner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems
- Computer Science, MathematicsISSAC
- 2019
This work introduces the first algorithm that overcomes the restriction of sparsity in the case where all the polynomials have the same sparsity structure, that is the same Newton polytope.
Differential forms in computational algebraic geometry
- Computer Science, MathematicsISSAC '07
- 2007
A randomised algorithm solving #ICC for a fixed number of rational equations given by straight-line programs (slps), which runs in parallel polylogarithmic time in the length and the degree of the slps.
Counting complexity classes for numeric computations II: algebraic and semialgebraic sets
- Mathematics, Computer ScienceSTOC '04
- 2004
It is proved that, for all k ∈ N, the FPSPACE-hardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial holds with respect to the singular homology as well as for the Borel-Moore homology.
Gröbner Bases and Nullstellensätze for Graph-Coloring Ideals
- Mathematics, Computer ScienceArXiv
- 2014
This paper provides lower bounds on the difficulty of computing Gr\"obner bases and Nullstellensatz certificates for the coloring ideals of general graphs and provides a polynomial-time algorithm for chordal graphs.
References
SHOWING 1-10 OF 93 REFERENCES
Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory
- Mathematics
- 1995
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…
The Structure of Polynomial Ideals and Gröbner Bases
- MathematicsSIAM J. Comput.
- 1990
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.
On the complexity of computing a Gröbner basis for the radical of a zero dimensional ideal
- MathematicsSTOC '90
- 1990
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
- MathematicsSTACS
- 1989
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
- MathematicsISSAC '96
- 1996
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
- Mathematics, Computer Science
- 1996
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
- Mathematics, Computer Science
- 1991
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
- Mathematics, Computer ScienceISSAC '96
- 1996
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
- Mathematics
- 1994
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
- MathematicsICALP
- 1996
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.