Truncated Plethystic Exponentials Preserve Power Sum Constraints

Abstract

Given an arbitrary sequence $(α_1, \ldots, α_n) \in \mathbb{C}^n$, we show that the degree-$n$ truncation of the formal exponential $\exp\bigl(-\sum_{k=1}^{\infty} \frac{α_k}{k} x^k\bigr)$ produces a polynomial whose roots $ρ_1, \ldots, ρ_n$ satisfy $\sum_{i=1}^n ρ_i^{-k} = α_k$ 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 $α_k = k^{1-s}$, 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 $\lim_{n\to\infty} P_n^{(s)}(1) = \exp(-ζ(s))$ for $\mathrm{Re}(s) > 1$.

Figures (1)

• Figure 1: (a) Roots of $P_n(x) = [\exp(-x/(1-x))]_{\deg \leq n}$ in the complex plane for $n = 2, \ldots, 50$, colored by $n$ (purple: small $n$, yellow: $n = 50$). (b) Heatmap of $\log_{10}|p_{-k}^{(n)} - \alpha_k|$ for $\alpha_k = k$, $n, k = 1, \ldots, 120$. White pixels (lower-left triangle, $k \leq n$) indicate errors below machine precision ($< 10^{-8}$), confirming Theorem \ref{['thm:main']}.

Theorems & Definitions (12)

• Theorem 1: Truncation-Exactness
• Definition 2: Log-generating function
• Definition 3: Associated exponential and truncated polynomial
• Lemma 4: Factored form
• proof
• Lemma 5: Logarithm of factored form
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Proposition 6: Coefficient recurrence
• proof