Sequences of fields with many solutions to the unit equation
Let \(K\) be an algebraic number field of degree \(\delta\), unit rank \(r\), with unit group \(U\), and denote by \(N\) the number of solutions of the unit equation \( x+y=1 (x,y\in U)\). \textit{J.-H. Evertse} [Invent. Math. 75, 561-584 (1984; Zbl 0521.10015)] has shown that \(N\leq 3\cdot 7^{\delta +2r}\). On the other hand \textit{T. Nagell} [Ark. Mat. 8, 115-127 (1970; Zbl 0213.06902)]\ proved that for \(\delta\geq 5\) there are infinitely many number fields \(K\) such that \(N\geq 6(2\delta -3)\). The purpose of the present paper is to fill the gap between the above mentioned upper and lower bounds. The author constructs sequences of fields for which \(N\) grows quadratically with \(\delta\).
scientific article; zbMATH DE number 1026400
scientific article