Diophantine equations with truncated binomial polynomials (Q904189)

The authors consider the Diophantine equation \(P_{n,k}(x)=P_{m,\ell}(y)\) in integers \(x,y\), where, for positive integers \(k\leq n\), \(P_{n,k}(x)=\sum_{j=0}^k {n\choose j}x^j\) is a truncated binomial expression of \((1+x)^n\). Under certain irreducibility assumptions, they show that the equation admits only finitely many solutions. There proof mainly relies on the Bilu-Tichy theorem. Further, they give some results concerning the functional decomposition of \(P_{n,k}(x)\) into two lower degree polynomials.