Izvorni znanstveni članak

https://doi.org/10.3336/gm.60.1.01

On a conjecture of Levesque and Waldschmidt

Tobias Hilgart orcid.org/0000-0002-5644-4749 ; Department of Mathematics, University of Salzburg, 5020 Salzburg, Austria
Volker Ziegler ; Department of Mathematics, University of Salzburg, 5020 Salzburg, Austria

Preuzmi JATS datoteku

Sažetak

One of the first parametrised Thue equations,
\[
\hspace{-18ex}\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1,
\]
over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as
\[
\hspace{-18ex} \left| N_{K/\mathbb{Q}}\left( X - \lambda_0 Y \right) \right| = \left| \left( X-\lambda_0 Y \right) \left( X-\lambda_1 Y \right) \left( X-\lambda_2 Y \right) \right| =1
\]
if \(\lambda_0, \lambda_1, \lambda_2\) are the roots of the defining irreducible polynomial, and \(K\) is the corresponding number field.
Levesque and Waldschmidt twisted this norm-form equation by an exponential parameter \(s\) and looked, among other things, at the equation
\[
\hspace{-18ex} \left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s Y \right) \right| = 1.
\]
They solved this effectively and conjectured that introducing a second exponential parameter \(t\) and looking at \(\left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s\lambda_1^t Y \right) \right| = 1\) does not change the effective solvability.
We want to partially confirm this if
\[
\hspace{-18ex} \min\left\{ \left| 2s-t \right|, \left| 2t-s \right|, \left| s+t \right| \right\} \gt \varepsilon \cdot \max\left\{ \left|s\right|, \left|t\right| \right\} \gt 2,
\]
i.e., the two exponents do not almost cancel in specific cases.

Ključne riječi

Parametrised Thue equations, exponential Diophantine equations

Hrčak ID:

332470

URI

https://hrcak.srce.hr/332470

Datum izdavanja:

10.3.2026.

Posjeta: 293 *