Polynomials determined by a few of their coefficients.
Let \(K\) be a field of characteristic zero. The authors consider the following Problem: Is it true that a monic polynomial \(f\in k[x]\) of degree \(n\) with exactly \(k\) distinct zeros is determined up to finitely many possibilities by any \(k\) of its non-zero proper coefficients? Here a coefficient is called proper if it is the coefficient of \(x^i\) for \(i<n\) and the assignment of \(k\) non-zero proper coefficients is to be meant as the assigment of \(k\) pairs \((i,a_i)\) where \(0\leq i<n\) and \(0\neq a_i\in K\). The answer is positive in a number of cases. To give an example, Theorem 1 of the paper states that if \(k=2\) the polynomial is determined up to \(n(n-1)[n/2]\) possibilities. Restricting to the case \(K={\mathbb Q}\), the authors prove as an application the finiteness, under suitable conditions, of the solutions \((x,y)\in {\mathbb Z}^2\) of the superelliptic Diophantine equation \(f(x)+g(x) = by^l\), where \(f,g\in{\mathbb Q}[x]\), \(b\in {\mathbb Q}\setminus \{0\}\) and \(l\) is an integer \(\geq 2\).
scientific article; zbMATH DE number 2124639