Bruce Reznick: Linear dependence among powers of polynomials

Abstract:
One variation of the Waring problem is to ask for the shortest non-trivial equations of the form \(f_1^d + ... + f_r^d = 0\), under various conditions on r, d and where \(f_j\) is a binary form. In this talk I'll limit myself to quadratic forms, and show all solutions for r=4 and d=3,4,5. I'll also give tools for you to find such equations on your own. The talk will touch on topics from algebra, analysis, number theory, combinatorics and algebraic geometry and name-check such notables as Euler, Sylvester and Ramanujan, but be basically self-contained. To whet your appetite:

\((x^2 + x y - y^2)^3 + (x^2 - x y - y^2)^3 = 2x^6 - 2y^6 \)