Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA, the VEGAS group at LO-RIA and the MPI-Saarbrücken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5 000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8 000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no "best method" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.

Abstract. In 1997, Bousquet-Mélou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the formWe continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n ≥ 1 if and only if s is an ℓ-sequence, i.e., recursively defined through s0 = 0, s1 = 1, and si = ℓsi−1 − si−2 for i ≥ 2. One consequence is that among such sequences s, unless s is an ℓ-sequence, the generating function for the s-lecture hall partitions can have the form ((1 − q e 1 )(1 − q e 2 ) · · · (1 − q en )) −1 for at most finitely many n.We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h * -vectors for s-lecture hall polytopes. We end with open questions and directions for further research.

Abstract. We investigate the arithmetic-geometric structure of the lecture hall coneWe show that Ln is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h * -polynomial is given by the (n − 1)st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for Ln, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of Ln, including connections between enumerative and algebraic properties of Ln and cones over unit cubes.

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 Omega combines 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 Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.1 arXiv:1501.07773v1 [math.CO] 30 Jan 2015In general, we represent a set S ⊂ Z n 0 of non-negative integer vectors by the multivariate generating functionIt is a well-known fact that when S is the set of solutions to a linear Diophantine system, then φ S is always a rational function ρ S . It is this rational function ρ S that we seek to compute when solving a given linear Diophantine system. We are not going to be interested in the normal form of ρ S , though, since the normal form may be unnecessarily large. Take for example the system x 1 + x 2 = 100. Here, the set of solutions is S = {(100, 0), (99, 1), . . . , (0, 100)} which can be represented by the rational functionNote that the polynomial on the right-hand side is the normal form of this rational function, which has 101 terms. (In fact, the normal form of a rational function representing a finite set will always be a polynomial.) The rational function expression on the left-hand side is not in normal form since the denominator divides the numerator, however it is much shorter, having only 4 terms. We are therefore interested in computing multivariate rational function expressions, which are not uniquely determined, as opposed to the unique rational function in normal form. Given this terminology and notation, we can now state the computational problem of solving linear Diophantine systems that this article is about. Problem 1.1. Rational Function Solution of Linear Diophantine System (rfsLDS)Output: An expression for the rational function ρ ∈ Q(z 1 , . . . , z n ) representing the set of all non-negative integer vectors x ∈ Z n 0 such that Ax b. Such rational function solutions to LDS are of great importance in many applications. For example, they can be used to prove theorems in number theory and combinatorics [4,5,50...

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