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Free semigroups of power series
Abstract
Given power series f1,…,fr∈ z2K[[z]] over a field K, we show that the semigroup ⟨ f1,…,fr⟩ generated by the f's under composition is free of rank r whenever the multiplicative semigroup ⟨lc(f1),…,lc(fr)⟩ in K× is free commutative of rank r; here lc(f) denotes the first nonzero coefficient of f. To do this, we associate an affine linear transformation to every F∈⟨ f1,…,fr⟩ and then apply a ping-pong lemma. In particular, combining this with work of Pappalardi, Sha, Shparlinski, and Stewart, it follows that most semigroups of polynomials over a number field K are free, and we make this statement precise through a counting argument.
Related
- Kobayashi's problem for power semigroups of inverse semigroups
- Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups
- Finite bases for full power semirings of finite nilpotent semigroups
- A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups
- Irreducibility of Endomorphisms of Finitely Generated Free Semigroups
BibTeX
@misc{hindes2026free,
title = {{Free semigroups of power series}},
author = {Wade Hindes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31582},
eprint = {2609.31582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31582},
}
RIS
TY - UNPB
TI - Free semigroups of power series
AU - Wade Hindes
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.31582
UR - https://arxiv.org/abs/2609.31582
AB - Given power series f_1,…,f_r∈ z^2K[[z]] over a field K, we show that the semigroup ⟨ f_1,…,f_r⟩ generated by the f's under composition is free of rank r whenever the multiplicative semigroup ⟨lc(f_1),…,lc(f_r)⟩ in K^× is free commutative of rank r; here lc(f) denotes the first nonzero coefficient of f. To do this, we associate an affine linear transformation to every F∈⟨ f_1,…,f_r⟩ and then apply a ping-pong lemma. In particular, combining this with work of Pappalardi, Sha, Shparlinski, and Stewart, it follows that most semigroups of polynomials over a number field K are free, and we make this statement precise through a counting argument.
ER -