Counting solutions to binomial complete intersections

Authors:
Eduardo Cattani;Alicia Dickenstein
Affiliations:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA;Departamento de Matematica, FCEyN, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina
Venue:
Journal of Complexity
Year:
2007

Citing 7
Cited 3

An optimal algorithm for constructing the reduced Gröbner basis of binomial ideals

Journal of Symbolic Computation
Primary decomposition of lattice basis ideals

Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
An Implementation of Tarjan's Algorithm for the Block Triangularization of a Matrix

ACM Transactions on Mathematical Software (TOMS)
An optimal algorithm for constructing the reduced Gröbner basis of binomial ideals, and applications to commutative semigroups

Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the second Magma conference
Elimination theory in codimension 2

Journal of Symbolic Computation
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
A Singular Introduction to Commutative Algebra

A Singular Introduction to Commutative Algebra

Complexity of solving systems with few independent monomials and applications to mass-action kinetics

CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Affine solution sets of sparse polynomial systems

Journal of Symbolic Computation
Computing Puiseux series for algebraic surfaces

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

Quantified Score

Hi-index	0.01

Visualization

Abstract

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.