Ehrhart Positivity of Tesler Polytopes and Berline–Vergne’s Valuation

For a ∈ R ≥ 0 n , the Tesler polytope Tes n ( a ) is the set of upper triangular matrices with non-negative entries whose hook sum vector is a . Motivated by a conjecture of Morales, we study the questions of whether the coefficients of the Ehrhart polynomial of Tes n ( 1 , 1 , ⋯ , 1 ) are positive. We attack this problem by studying a certain function constructed by Berline–Vergne and its values on faces of a unimodularly equivalent copy of Tes n ( 1 , 1 , ⋯ , 1 ) . We develop a method of obtaining the dot products appeared in formulas for computing Berline–Vergne’s function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline–Vergne’s function has positive values on codimension 2 and 3 faces of the polytopes we consider. As a consequence, we prove that the third and fourth coefficients of the Ehrhart polynomial of Tes n ( 1 , ⋯ , 1 ) are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of Tes n ( 1 , ⋯ , 1 ) including all the integral Tesler polytopes.