Binomial Thue's equation over function fields

Julia Mueller

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It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_{t} (x) = x^{4} - t x^{3} - 6 x^{2} + t x + 1$ , assuming that $t > 0$ , $t \neq 3$ and $t^{2} + 16$ has no odd square factors. In addition to generators of power integral bases we also calculate...

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We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. " $d$ -th root problem"), which in short asks whether the integrality of the values of the quotient (resp. $d$ -th root) of two (resp. one) linear recurrences implies that this quotient (resp. $d$ -th root) is itself a recurrence. We shall also...