A simultaneous approximation problem for exponentials and logarithms
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915307251040256 |
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| author | Kumar, Veekesh Tosi, Riccardo |
| author_facet | Kumar, Veekesh Tosi, Riccardo |
| contents | Let $α_1,α_2$ be non-zero algebraic numbers such that $\frac{\log α_2}{\logα_1}\notin\mathbb{Q}$ and let $β$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials $P(x,y,z)$ and $Q(x,y,z)$ with integer coefficients are not too small at the point $\left(\frac{\logα_2}{\log α_1},α_1^β, α_2^β\right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\logα_2}{\log α_1}$, $α^β_1$ and $α^β_2$ which are algebraically independent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20957 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A simultaneous approximation problem for exponentials and logarithms Kumar, Veekesh Tosi, Riccardo Number Theory Let $α_1,α_2$ be non-zero algebraic numbers such that $\frac{\log α_2}{\logα_1}\notin\mathbb{Q}$ and let $β$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials $P(x,y,z)$ and $Q(x,y,z)$ with integer coefficients are not too small at the point $\left(\frac{\logα_2}{\log α_1},α_1^β, α_2^β\right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\logα_2}{\log α_1}$, $α^β_1$ and $α^β_2$ which are algebraically independent. |
| title | A simultaneous approximation problem for exponentials and logarithms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.20957 |