Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

Abstract

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...

The Floquet Theory of the Two-Level System Revisited

“…where the n in the last term denotes the winding number of γ around the z-axis. This result is in close analogy to equation (2.35) of [26] that represents the semi-classical limit of the quasienergy for integrable Floquet systems. It thus seems that for the two level system, similar as in the case of the driven harmonic oscillator, the semiclassical limit of the quasienergy and the exact quantumtheoretical expression coincide.…”

“…The classical mechanics approach in subsection III D shows that the quasienergy is essentially the integral of the Poincaré-Cartan form over one closed orbit. This result is closely connected to the approach to semiclassical Floquet theory in [26].…”

“…Especially, the quasienergy is essentially given by the action integral over one period of the classical solution. This is reminiscent of the semiclassical Floquet theory developed in [26]. In the special case of the Rabi problem with elliptical polarization our approach yields the result that Shirley's resonance condition is equivalent to the vanishing of the time average of the component of the classical periodic solution into the direction of the constant magnetic field.…”

“…The correspondence between T -periodic solutions X(t) of (42) and Floquet solutions ψ(t) of (1) that has been formulated in Assertion 1 can be further analyzed w. r. t. two different geometric perspectives: Either the map ψ(t) → X(t) can be viewed as the restriction of the map of a Hilbert space H onto the corresponding projective Hilbert space P (H) and the quasienergy as a phase change during a cyclic quantum evolution in the sense of [27]. Or the Bloch sphere can be construed as the phase space of the classical limit of the two level system and the quasienergy can be related to its semi-classical limit in the sense of [26]. We will treat both aspects in the following subsections.…”

“…It thus seems that for the two level system, similar as in the case of the driven harmonic oscillator, the semiclassical limit of the quasienergy and the exact quantumtheoretical expression coincide. However, it has not yet been shown that the quantization procedure adopted in [26] yields the quantum two level system when starting from its classical limit.…”