Pseudo-Effectivity in the Skolem-Mahler-Lech Theorem
Abstract
The Skolem--Mahler--Lech Theorem asserts that a non-degenerate linear recurrence sequence (LRS) has only finitely many zeros, but is ineffective in the sense that no general procedure is known to determine them. In this paper a \emph{pseudo-effective} approach is introduced: rather than seeking effective bounds for every sequence, one obtains explicit bounds valid for all but a controlled number of sequences of bounded height, up to a natural equivalence. Such results are established for order $3$ LRS's, and extended to all higher orders subject to suitable height constraints on the characteristic roots. The underlying mechanism is a quantitative theory for exponential-polynomial equations subject to low-height multiplicative perturbations. In particular, it is shown that the finiteness phenomenon underlying the Skolem--Mahler--Lech Theorem persists under arithmetic perturbations whose height is permitted to grow linearly at a sufficiently small rate. As a further application, a moving-target finiteness result is established for linear orbits entering expanding arithmetic neighbourhoods of proper linear subspaces.
Related
- Skolem-Mahler-Lech in rings of positive characteristic: a shorter proof and a multi-dimensional generalization
- IC-rigidity and generalizations of a theorem of Mahler
- Mahler Method and Vanishing Theorem
- Hard Lefschetz theorem for logarithmic sheaves twisted by pseudo-effective line bundles
- Finite-tower bounds for Skolem functions
BibTeX
@misc{shirandami2026pseudoeffectivity,
title = {{Pseudo-Effectivity in the Skolem-Mahler-Lech Theorem}},
author = {Victor Shirandami},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32901},
eprint = {2609.32901},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32901},
}
RIS
TY - UNPB
TI - Pseudo-Effectivity in the Skolem-Mahler-Lech Theorem
AU - Victor Shirandami
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32901
UR - https://arxiv.org/abs/2609.32901
AB - The Skolem--Mahler--Lech Theorem asserts that a non-degenerate linear recurrence sequence (LRS) has only finitely many zeros, but is ineffective in the sense that no general procedure is known to determine them. In this paper a \emph{pseudo-effective} approach is introduced: rather than seeking effective bounds for every sequence, one obtains explicit bounds valid for all but a controlled number of sequences of bounded height, up to a natural equivalence. Such results are established for order $3$ LRS's, and extended to all higher orders subject to suitable height constraints on the characteristic roots. The underlying mechanism is a quantitative theory for exponential-polynomial equations subject to low-height multiplicative perturbations. In particular, it is shown that the finiteness phenomenon underlying the Skolem--Mahler--Lech Theorem persists under arithmetic perturbations whose height is permitted to grow linearly at a sufficiently small rate. As a further application, a moving-target finiteness result is established for linear orbits entering expanding arithmetic neighbourhoods of proper linear subspaces.
ER -