# Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

@article{Breuer2015PolyhedralOA, title={Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems}, author={Felix Breuer and Zafeirakis Zafeirakopoulos}, journal={Annals of Combinatorics}, year={2015}, volume={21}, pages={211-280} }

Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omegacombines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as…

## 10 Citations

### Matrix computations with the Omega calculus

- Mathematics, Computer ScienceLinear and Multilinear Algebra
- 2021

An extension of the Omega calculus in the context of matrix analysis introduced recently by Neto is explored, obtaining Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular.

### Computing efficiently the non-properness set of polynomial maps on the plane

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- 2021

The algorithm takes into account the sparsity of polynomials as it depends on (the Minkowski sum of) the Newton polytopes of f and provides a finer representation of the set of non-properness as a union of algebraic or semi-algebraic sets that correspond to edges of the Newtonpolytope, which is of independent interest.

### A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

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- 2015

The smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied…

### Partition analysis, modular functions, and computer algebra

- Mathematics
- 2016

This article describes recent developments connecting problems of enumerative combinatorics, constrained by linear systems of Diophantine inequalities, with number theory topics like partitions,…

### Generating Functions of Weighted Voting Games, MacMahon's Partition Analysis, and Clifford Algebras

- MathematicsMath. Oper. Res.
- 2019

A new generalized generating function is introduced that gives new generating functions that give, for fixed coalitions, all the distribution of weights of the players of the voting game such that a given player swings or not.

### Interview with Peter Paule

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- 2022

Peter Paule obtained a Ph.D. from the University of Vienna in 1982 under the supervision of Johann Cigler. In 1996 he earned a habilitation from Johannes Kepler University. Since 1990 he has held a…

### Matrix Analysis and Omega Calculus

- Computer Science, MathematicsSIAM Rev.
- 2020

A new operator based approach to matrix analysis is introduced that is an extension of a tool introduced long ago by MacMahon to analyze the partitions of partitions in matrix analysis.

### Plane partitions in the work of Richard Stanley and his school

- Art
- 2015

These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.

### Fixed-point iterative approach for solving linear Diophantine systems with bounds on the variables

- MathematicsCyber-Physical Systems
- 2022

### An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

- MathematicsComputer Algebra and Polynomials
- 2015

In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics…

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