Progress on the unfair 0-1-polynomials conjecture using linear recurrences and numerical analysis

Abstract: If the product of two monic polynomials with real nonnegative coefficients has all coefficients equal to 0 or 1, does it follow that all the coefficients of the two factors are also equal to 0 or 1? Here is an equivalent formulation of this intriguing problem: is it possible to weigh unfairly a pair of dice so that the probabilities of every possible outcome (roll them and take the sum) were the same? If the two dice have six faces numbered 1 to 6, it is easy to show that the answer is no. But for general dice with finitely many faces, this is an open problem with no significant advancement since 1937. In this paper we examine, in some sense, the first infinite family of cases that cannot be treated with classical methods: the first die has three faces numbered with 0,2 and 5, while the second die is arbitrary. In other words, we examine factorizations of 0-1-polynomials with one factor equal to $x^5+a x^2 +1$ for some nonnegative $a$. We discover that this case may be solved (that is, necessarily $a=0$ or $a=1$) using the theory of linear recurrence sequences, computation of resultants, a fair amount of analytic and numerical approximations... and a little bit of luck.