Pure product polynomials and the Prouhet-Tarry-Escott problem

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

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