Polynomial maps on the monoid of words

Abstract

We briefly visit the theory of polynomial and semipolynomial maps defined on an arbitrary monoid, with range a commutative group. Then we characterize the space 𝒫(S,ℂ) of polynomial maps f:S→ ℂ, where S=𝒜^* is the monoid of words based on an arbitrary alphabet 𝒜 under concatenation, and we use this characterization to prove that if there exists a monoid Snot∈𝒞𝒮 such that 𝒮𝒫(S,ℂ)≠ 𝒫(S,ℂ), then also 𝒮𝒫(𝒜^*,ℂ)≠𝒫(𝒜^*,ℂ) for a certain alphabet 𝒜. We propose as an open problem to prove or disprove that 𝒮𝒫(𝒜^*,ℂ)=𝒫(𝒜^*,ℂ) for arbitrary alphabets 𝒜. Our results are motivated by previous work of Shulman.

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BibTeX RIS

BibTeX

@misc{almira2026polynomial,
  title = {{Polynomial maps on the monoid of words}},
  author = {J. M. Almira},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.05077},
  eprint = {2606.05077},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.05077},
}

RIS

TY  - UNPB
TI  - Polynomial maps on the monoid of words
AU  - J. M. Almira
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2606.05077
UR  - https://arxiv.org/abs/2606.05077
AB  - We briefly visit the theory of polynomial and semipolynomial maps defined on an arbitrary monoid, with range a commutative group. Then we characterize the space 𝒫(S,ℂ) of polynomial maps f:S→ ℂ, where S=𝒜^* is the monoid of words based on an arbitrary alphabet 𝒜 under concatenation, and we use this characterization to prove that if there exists a monoid Snot∈𝒞𝒮 such that 𝒮𝒫(S,ℂ)≠ 𝒫(S,ℂ), then also 𝒮𝒫(𝒜^*,ℂ)≠𝒫(𝒜^*,ℂ) for a certain alphabet 𝒜. We propose as an open problem to prove or disprove that 𝒮𝒫(𝒜^*,ℂ)=𝒫(𝒜^*,ℂ) for arbitrary alphabets 𝒜. Our results are motivated by previous work of Shulman.
ER  -

Abstract

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