The Erdős—Szekeres Problem

Abstract

One approach to the Prouhet-Tarry-Escott problem is to construct products of the form

$$ p(z): = \prod\limits_{{k = 1}}^N {\left( {1 - {z^{{{\alpha_i}}}}} \right)} $$

. This product has a zero of order N at 1, and the idea is to try to minimize the length (the l ₁ norm) of p. We denote by E ^* _N the minimum possible l ₁ norm of any TV-term product of the above form. The l ₁ norm is just the sum of the absolute values of the coefficients of the polynomial p when it is expanded, and an ideal solution of the Prouhet-Tarry-Escott problem arises when E ^* _N = 2N (as in Theorem 1(c) of Chapter 11).