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Algebraic independence of infinite series

Jaroslav Hancl, Mathias L. Laursen, Simon Kristensen

TL;DR

This work develops criteria ensuring algebraic independence for finite families of numbers given by rational-series, extending Erdős–Erdős irrationality results to the multivariate algebraic setting. By combining $p$-adic divisibility patterns with a unifying size-sequence and, in the two-series case, projective-curve genus arguments, the authors prove that the vector $(\alpha_1,\dots,\alpha_K)$ cannot satisfy any non-zero polynomial of bounded degree. The framework yields corollaries on $p$-irrationality, $p$-transcendence, and more general $\mathcal{P}$-irrationality/transcendence, and is illustrated with numerous concrete examples including sign-varying series and zeta-type sums. The results significantly broaden irrationality and independence criteria for series with rational terms, with potential applications to explicit constructions and Diophantine problems. The methods blend analytic tail estimates with algebraic-geometry tools (genus, Faltings’s theorem), providing both qualitative and quantitative criteria for independence.

Abstract

We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and $mathbb{Q}$-linear independence of such series.

Algebraic independence of infinite series

TL;DR

This work develops criteria ensuring algebraic independence for finite families of numbers given by rational-series, extending Erdős–Erdős irrationality results to the multivariate algebraic setting. By combining -adic divisibility patterns with a unifying size-sequence and, in the two-series case, projective-curve genus arguments, the authors prove that the vector cannot satisfy any non-zero polynomial of bounded degree. The framework yields corollaries on -irrationality, -transcendence, and more general -irrationality/transcendence, and is illustrated with numerous concrete examples including sign-varying series and zeta-type sums. The results significantly broaden irrationality and independence criteria for series with rational terms, with potential applications to explicit constructions and Diophantine problems. The methods blend analytic tail estimates with algebraic-geometry tools (genus, Faltings’s theorem), providing both qualitative and quantitative criteria for independence.

Abstract

We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and -linear independence of such series.

Paper Structure

This paper contains 5 sections, 22 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples and applications
  4. Preliminaries
  5. Proof of Theorems \ref{['thm:general']} and \ref{['thm:general infinite']}

Key Result

Theorem 1

Let $K$ and $d$ be positive integers, let $\varepsilon$ and $\kappa$ be positive real numbers with $\kappa <1$, and let $p$ be a prime number. For $k=1,\ldots, K$, let $\{a_{k,n}\}_{n=1}^\infty$ and $\{b_{k,n}\}_{n=1}^\infty$ be sequences of non-zero integers with $p\nmid \gcd(a_{k,n},b_{k,n})$ such for exactly one $n\le N$ and writing $a_{0,n}=1$ for all $n$. Suppose there is a sequence $\{a_n\}

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • proof
  • Remark 1
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 39 more