### Abstract

We give a variant of Barvinok's algorithm for computing a short rational generating function for the integer points in P := {x ∈ R^{n} : Ax ≤ b}; a use of which is to count the number of integer points in P. We use an algebraic perturbation, replacing each b_{i} with b_{i}+τ^{i}, where τ>0 is an arbitrarily small indeterminate. Hence, our new right-hand vector has components in the ordered ring Q[τ] of polynomials in τ. Denoting the perturbed polyhedron by P(τ)⊂R[τ]^{n}, we use the facts that: P(τ) is full dimensional, simple, and contains the same integer points as P.

Original language | English (US) |
---|---|

Pages (from-to) | 15-20 |

Number of pages | 6 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 50 |

DOIs | |

State | Published - Dec 1 2015 |

### Fingerprint

### Keywords

- Barvinok's algorithm
- Generating function
- Integer
- Polyhedron

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*50*, 15-20. https://doi.org/10.1016/j.endm.2015.07.004

**An algebraic-perturbation variant of Barvinok's algorithm.** / Lee, Jon; Skipper, Daphne E.

Research output: Contribution to journal › Article

*Electronic Notes in Discrete Mathematics*, vol. 50, pp. 15-20. https://doi.org/10.1016/j.endm.2015.07.004

}

TY - JOUR

T1 - An algebraic-perturbation variant of Barvinok's algorithm

AU - Lee, Jon

AU - Skipper, Daphne E

PY - 2015/12/1

Y1 - 2015/12/1

N2 - We give a variant of Barvinok's algorithm for computing a short rational generating function for the integer points in P := {x ∈ Rn : Ax ≤ b}; a use of which is to count the number of integer points in P. We use an algebraic perturbation, replacing each bi with bi+τi, where τ>0 is an arbitrarily small indeterminate. Hence, our new right-hand vector has components in the ordered ring Q[τ] of polynomials in τ. Denoting the perturbed polyhedron by P(τ)⊂R[τ]n, we use the facts that: P(τ) is full dimensional, simple, and contains the same integer points as P.

AB - We give a variant of Barvinok's algorithm for computing a short rational generating function for the integer points in P := {x ∈ Rn : Ax ≤ b}; a use of which is to count the number of integer points in P. We use an algebraic perturbation, replacing each bi with bi+τi, where τ>0 is an arbitrarily small indeterminate. Hence, our new right-hand vector has components in the ordered ring Q[τ] of polynomials in τ. Denoting the perturbed polyhedron by P(τ)⊂R[τ]n, we use the facts that: P(τ) is full dimensional, simple, and contains the same integer points as P.

KW - Barvinok's algorithm

KW - Generating function

KW - Integer

KW - Polyhedron

UR - http://www.scopus.com/inward/record.url?scp=84953378005&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84953378005&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2015.07.004

DO - 10.1016/j.endm.2015.07.004

M3 - Article

VL - 50

SP - 15

EP - 20

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -