On the numbers of solutions of decomposable polynomial equations
In a recent paper \textit{J. H. Evertse} and \textit{K. Györy} [Acta Arith. 50, No.4, 357-379 (1988; Zbl 0595.10013)] gave criteria for the finiteness of the numbers of solutions of decomposable form equations (including Thue equations, norm form equations, discriminant form equations and index form equations) over finitely generated integral domains over \({\mathbb{Z}}\). On the other hand \textit{I. Gaál} considered inhomogeneous extensions of certain classes of decomposable form equations [cf. Publ. Math. 34, 109-122 (1987; Zbl 0626.10015)]. In the present paper the authors give criteria for the finiteness of the numbers of solutions of decomposable polynomial equations (i.e. ``inhomogeneous'' decomposable form equations) over fis S(d,c): For each continuous function f on [-1,1]\(\times ({\mathbb{R}}/{\mathbb{Z}})\) \[ \lim_{N\to \infty}(1/N)\sum^{N}_{c=1}(1/c)\sum_{d mod c, (d,c)=1}f(S(d,c)/c, d/c)=(6/\pi^ 2)\int^{1}_{0}f(0,t) dt \] and \[ \lim_{N\to \infty}(1/N)\sum^{N}_{c=1}\sum_{d mod c, (d,c)=1}(S(d,c)/c)^ 2 f(S(d,c)/c, d/c)= \] \[ \sum^{\infty}_{n=1}\sum_{d| n}\beta (n/d)(1/d)\sum_{m mod d}(1/12n)^ 2\{f(1/12n, m/d)+f(-1/12n, m/d)\} \] with \(\beta\) (k) defined by \(\sum^{\infty}_{k=1}\beta (k)k^{- s}=\zeta (s) \zeta (s+1)^{-2}\) for Re s\(>1\).
scientific article; zbMATH DE number 4068977