Residues formulae for volumes and Ehrhart polynomials of convex polytopes.

By W. Baldoni-Silva, Michèle Vergne. 2001

61 citations recorded by the source corpus.

In these notes, we explain residue formulae for volumes of convex polytopes, and for Ehrahrt polynomials based on the notion of total residue. We apply this method to the computation of the volume of the Chan-Robbins polytope. The final computation is based on a total residue formula for the system $A_n$, similar to Morris identity. For flow polytopes, a formula of change of variables in total residues leads to a "nice formula" for Ehrhart polynomials in function of mixed volumes. We apply it to Pitman-Stanley polytope.

Subfields: Algebra and Number Theory, Applied Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/49a65512900a559b38c8d28f2b3258c10ed310cf

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In Algebra and Number Theory

In Applied Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Flow polytopes of signed graphs and the Kostant partition function (2012, methodology): “Baldoni and Vergne use residue techniques, while in their unpublished work Postnikov and Stanley took a combinatorial approach [19, 20].”
Kostant Partitions Functions and Flow Polytopes (2008, methodology): “The ideas of this section are valid for general convex polytopes [4], [24], thus we describe first the general setting.”
An Invitation to the Generalized Saturation Conjecture (2004, methodology): “By definition, the inequality P1(q) ≥ P2(q) means that the difference P1(q)−P2(q) is a polynomial with non–negative real coefficients.”
Effective lattice point counting in rational convex polytopes (2004, methodology): “We also tested the performance in the case of two-way contingency tables and Kostant’s partition function where special purpose software has already been written (Baldoni-Silva and Vergne, 2001; Beck and Pixton, 2003; De Loera and Sturmfels, 2003; Mount, 2000).”