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Tobias Hilgart

Publications and source records attributed to Tobias Hilgart.

8 recordsLinked to original sources
arXiv2026

Complete Resolution Of A Family Of Twisted Thue Equations

One of the first infinite families of Thue equations, $$F_n(X)=X^3 - (n-1) X^2Y - (n+2)XY^2 - Y^3 = \pm 1$$ for $n\in \mathbb{Z}$, was solved by Thomas in 1990. This family is associated to the simplest cubic fields $\mathbb{Q}(λ)$ of Shanks, where $λ$ is a root of $F_n(X,1)$. Levesque and Waldschmidt twisted the Thue equations by an exponent $t$ and looked at the equation $$N_{\mathbb{Q}(λ)/\mathbb{Q}}(X-λ^t Y)=\pm 1,$$ where $t\in \mathbb{Z}$ with $t\neq 0$. In this paper, we find all solutions $(X,Y,n,t)\in \mathbb{Z}^4$ with $n,t\in\mathbb{Z}$ and $t\neq 0$ to this family of twisted Thue equations, thereby answering a question of Levesque and Waldschmidt.

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arXiv2024

On split families of Thue equations with linear recurrence sequences as factors

We consider a parametrised family of Thue equations, \[ (x-G_1(n)\, y) \cdots (x-G_d(n)\, y) - y^d = \pm 1, \] which was first considered by Thomas to have an explicit set of solutions for parameters $n$ larger than some effectively computable constant. In the case where the parameter functions are polynomials belonging to an explicitly described family, this is known to be true. We consider other parameter functions, namely linear recurrence sequences, for which it is not obvious that a similar result holds, and confirm that it does for an explicitly described family of linear recurrence sequences.

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arXiv2024

On a conjecture of Levesque and Waldschmidt II

Related to Shank's notion of simplest cubic fields, the family of parametrised Diophantine equations, \[ x^3 - (n-1) x^2 y - (n+2) xy^2 - 1 = \left( x - λ_0 y\right) \left(x-λ_1 y\right) \left(x - λ_2 y\right) = \pm 1, \] was studied and solved effectively by Thomas and later solved completely by Mignotte. An open conjecture of Levesque and Waldschmidt states that taking these parametrised Diophantine equations and twisting them not only once but twice, in the sense that we look at \[ f_{n,s,t}(x,y) = \left( x - λ_0^s λ_1^t y \right) \left( x - λ_1^sλ_2^t y \right) \left( x - λ_2^sλ_0^t y \right) = \pm 1, \] retains a result similar to what Thomas obtained in the original or Levesque and Waldschidt in the once-twisted ($t = 0$) case; namely, that non-trivial solutions can only appear in equations where the parameters are small. We confirm this conjecture, given that the absolute values of the exponents $s, t$ are not too large compared to the base parameter $n$.

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arXiv2023

On a conjecture of Levesque and Waldschmidt

One of the first parametrised Thue equations, $$\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1,$$ over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as $$\left| N_{K/\mathbb{Q}}\left( X - λ_0 Y \right) \right| = \left| \left( X-λ_0 Y \right) \left( X-λ_1 Y \right) \left( X-λ_2 Y \right) \right| =1$$ if $λ_0, λ_1, λ_2$ are the roots of the defining irreducible polynomial, and $K$ the corresponding number field.\par\medskip Levesque and Waldschmidt twisted this norm-form equation by an exponential parameter $s$ and looked, among other things, at the equation $$\left| N_{K/\mathbb{Q}}\left( X - λ_0^s Y \right) \right| = 1.$$ They solved this effectively and conjectured that introducing a second exponential parameter $t$ and looking at $$\left| N_{K/\mathbb{Q}}\left( X - λ_0^sλ_1^t Y \right) \right| = 1$$ does not change the effective solvability. \par\medskip We want to partially confirm this, given that $$\min\left( \left| 2s-t \right|, \left| 2t-s \right|, \left| s+t \right| \right) > \varepsilon \cdot \max\left( \left|s\right|, \left|t\right| \right) > 2,$$ i.e. the two exponents do not almost cancel in specific cases.

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arXiv2023

Twisted Thue equations with multiple exponents in fixed number fields

Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $γ_1, \dots, γ_s \in K^*$ 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 \[ \left|N_{K/\mathbb{Q}}\left(X-γ_1^{t_1}\cdotsγ_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( γ_1^{t_1}\cdots γ_s^{t_s} \right) = K$, all of which are effectively computable.

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arXiv2022

On families of cubic split Thue equations parametrised by linear recurrence sequences

Let $(A_n)_{n\in \mathbb{N}}, (B_n)_{n\in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}$ be two linear-recurrent sequences that meet a dominant root condition and a few more technical requirements. We show that the split family of Thue equations \[ |X(X-A_n Y)(X-B_n Y) - Y^3| = 1 \] has but the trivial solutions $\pm\{ (0,1), (1,0), (A_n,1), (B_n,1) \}$, if the parameter $n$ is larger than some effectively computable constant.

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