Polynomial-exponential equations involving multi-recurrences

Abstract

In this paper we consider polynomial-exponential Diophantine equations of the form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$G_n^{(0)} y^d + G_n^{(1)} y^{d - 1} + \cdots + G_n^{(d - 1)} y + G_n^{(d)} = 0$$ \end{document} where Gn( i ) are multi-recurrences, i.e. polynomial-exponential functions in variables n = ( n1 ,..., nk ). Under suitable (but restrictive) conditions we prove that there are finitely many multi-recurrences Hn(1) ,..., Hn( s ) such that for all solutions ( n1 ,..., nk , y ) ∈ ℕ k × ℤ we either have \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_n^{(i)} = 0 or y = H_n^{(j)}$$ \end{document} for certain 1 ≦ i,j ≦ s , respectively. This generalizes earlier results of this type on such equations. The proof uses a recent result by Corvaja and Zannier.