Random walks through the areal Mahler measure: steps in the complex plane
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915217896636416 |
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| author | Lalín, Matilde N. Nair, Siva Sankar Ringeling, Berend Roy, Subham |
| author_facet | Lalín, Matilde N. Nair, Siva Sankar Ringeling, Berend Roy, Subham |
| contents | We study the areal Mahler measure of the two-variable, $k$-parameter family $x+y+k$ and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family $(x+1)(y+1)+kz$ in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random walks through the areal Mahler measure: steps in the complex plane Lalín, Matilde N. Nair, Siva Sankar Ringeling, Berend Roy, Subham Number Theory Complex Variables 11R06 (Primary) 11M06, 11R42 (Secondary) We study the areal Mahler measure of the two-variable, $k$-parameter family $x+y+k$ and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family $(x+1)(y+1)+kz$ in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family. |
| title | Random walks through the areal Mahler measure: steps in the complex plane |
| topic | Number Theory Complex Variables 11R06 (Primary) 11M06, 11R42 (Secondary) |
| url | https://arxiv.org/abs/2503.22536 |