On a property of the polynomials $x^n + (1-x)^n + a^n$

Abstract

The polynomials $x^n + (1-x)^n + a^n$ arise naturally from FLT (Fermat's Last Theorem). We formulate a conjecture about them which is a generalization of FLT. We investigate the complex roots of these polynomials, and our main result is that in the cases $|a|\leq \frac12$ and $a=-1$, they lie on an explicitly given curve while 'filling in' that curve. We hypothesize that this property can be generalized to hold for other $a$ as well.