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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.28828 |
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| _version_ | 1866914432776404992 |
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| author | Phalak, Yogesh |
| author_facet | Phalak, Yogesh |
| contents | 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 $\mathrm{Re}(s) > 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28828 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Truncated Plethystic Exponentials Preserve Power Sum Constraints Phalak, Yogesh Number Theory Combinatorics 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 $\mathrm{Re}(s) > 1$. |
| title | Truncated Plethystic Exponentials Preserve Power Sum Constraints |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2603.28828 |