Truncated Plethystic Exponentials Preserve Power Sum Constraints

TL;DR

The paper proves that truncating a specific formal exponential yields polynomials whose roots exactly match given power sum constraints, with applications to polylogarithms and the Riemann zeta function.

Contribution

It establishes an algebraic identity for truncations of formal exponentials, enabling exact root-power sum correspondence and efficient computation methods.

Findings

Truncated exponential polynomials satisfy exact power sum constraints.

The method provides an $O(n^2)$ algorithm for polynomial reconstruction.

Application to polylogarithms links to values of the Riemann zeta function.

Abstract

Given an arbitrary sequence $(\alpha_1, \ldots, \alpha_n) \in \mathbb{C}^n$, we show that the degree-$n$ truncation of the formal exponential $\exp\bigl(-\sum_{k=1}^{\infty} \frac{\alpha_k}{k} x^k\bigr)$ produces a polynomial whose roots $\rho_1, \ldots, \rho_n$ satisfy $\sum_{i=1}^n \rho_i^{-k} = \alpha_k$ exactly for $k = 1, \ldots, n$. This truncation-exactness property is an algebraic identity in the ring of formal power series, proved by coefficient matching. It defines a natural embedding of sequences into multisets of complex numbers and yields an $O(n^2)$ algorithm for computing the polynomial from the prescribed power sums. We apply the result to the polylogarithm family $\alpha_k = k^{1-s}$, where the associated exponential $\exp(-\mathrm{Li}_s(x))$ produces factorial-integer coefficient sequences for $s \leq 0$ and encodes values of the Riemann zeta function through $\lim_{n\to\infty} P_n^{(s)}(1) = \exp(-\zeta(s))$ for $\mathrm{Re}(s) > 1$.