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The Todd polynomials $td_k=td_k(b_1,b_2,\dots,b_m)$ are defined by theirgenerating functions $$\sum_{k\ge 0} td_k s^k = \prod_{i=1}^m \frac{b_is}{e^{b_i s}-1}.$$ It appears as a basic block in Todd class of a toricvariety, which is important in the theory of lattice polytopes and in numbertheory. We find generalized Todd polynomials arise naturally in MacMahon'spartition analysis, especially in Ehrhart series computation. We give fastevaluation of generalized Todd polynomials for numerical $b_i$'s. In order todo so, we develop fast operations in the quotient ring $\mathbb{Z}_p[[s]]$modulo $s^{d+1}$ for large prime $p$. As applications, i) we recompute theEhrhart series of magic squares of order 6, which was first solved by the firstnamed author. The running time is reduced from $70$ days to about $1$ day; ii)we give a polynomial time algorithm for Integer Linear Programming when thedimension is fixed, with a good performance.

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