Pure product polynomials and the Prouhet-Tarry-Escott problem

Abstract:

An $n$-factor pure product is a polynomial which can be expressed in the form $\prod _{i=1}^n(1-x^{\alpha _i})$ for some natural numbers $\alpha _1,\ldots ,\alpha _n$. We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every $n$-factor pure product has norm at least $2n$. We describe three algorithms for determining the least norm an $n$-factor pure product can have. We report results of our computations using one of these algorithms which include the result that every $n$-factor pure product has norm strictly greater than $2n$ if $n$ is $7$, $9$, $10$, or $11$.