Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming

TL;DR

This work presents a fast, modular arithmetic framework for evaluating generalized Todd polynomials and their constant terms, linking MacMahon’s partition analysis with Ehrhart theory and fixed-dimension ILP. The core strategy combines log-exponential tricks, FFT-accelerated manipulations in $\mathbb{Z}_p[[s]]/\langle s^d\rangle$, and carefully structured constant-term computations (CTGTodd) to achieve substantial speedups in Ehrhart series for problems like magic squares and in solving fixed-dimension ILP. Key contributions include explicit complexity bounds for seven algebraic operations, a fast algorithm for generalized Todd polynomials, and a practical Binary Search CT approach (BSCT) for ILP that runs in polynomial time when the dimension is fixed. The practical impact is demonstrated by dramatic reductions in computation time for Ehrhart series and by providing a polynomial-time ILP procedure in the fixed-dimension regime, with broad applicability to lattice-point counting and combinatorial enumeration problems.

Abstract

The Todd polynomials, denoted as $td_k(b_1,b_2,\ldots,b_m)$, are characterised by their generating functions: $$\sum_{k\ge 0} td_k s^k = \prod_{i=1}^m \frac{b_i s}{e^{b_i s}-1}.$$ These polynomials serve as fundamental components in the Todd class of toric varieties, a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalised Todd polynomials emerge naturally within the framework of MacMahon's partition analysis, particularly in the context of computing Ehrhart series. We introduce an efficient method for the evaluation of generalised Todd polynomials for numerical values of $b_i$. This is achieved through the development of expedited operations in the quotient ring $\mathbb{Z}_p[[s]]$ modulo $s^{d}$, where $p$ is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming when the dimension is fixed, exhibiting a notable enhancement in computational efficiency.

Figures (1)

• Figure 1: The polytope $P(k)$ and its new vertices $v_1$ and $v_2$.

Theorems & Definitions (54)

• Definition 1
• Proposition 2
• proof : Sketched proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6