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Twisted Thue equations with multiple exponents in fixed number fields
Number Theory · Submitted 13 Dec 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-\gamma_1^{t_1}\cdots\gamma_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( \gamma_1^{t_1}\cdots \gamma_s^{t_s} \right) = K$, all of which are effectively computable.
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The abstract, sentence by sentence
Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $\gamma_1, \dots, \gamma_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-\gamma_1^{t_1}\cdots\gamma_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( \gamma_1^{t_1}\cdots \gamma_s^{t_s} \right) = K$, all of which are effectively computable.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $\gamma_1, \dots, \gamma_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. | background 0.98 | 0.20 | 0.03 | asserted 0.84 |
| s1 | We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-\gamma_1^{t_1}\cdots\gamma_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( \gamma_1^{t_1}\cdots \gamma_s^{t_s} \right) = K$, all of which are effectively computable. | result 0.87 | 0.96 | 0.95 | asserted 1.00 |
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Original abstract
Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $\gamma_1, \dots, \gamma_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-\gamma_1^{t_1}\cdots\gamma_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( \gamma_1^{t_1}\cdots \gamma_s^{t_s} \right) = K$, all of which are effectively computable.
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