Linearly dependent powers of binary quadratic forms

Abstract

Given an integer $d\ge 2$, what is the least $r$ so that there is a set of binary quadratic forms $\{f_1,\dots ,f_r\}$ for which $\{f_j^d\}$ is non-trivially linearly dependent? We show that if $r\le 4$, then $d\le 5$, and for $d\ge 4$, construct such a set with $r=\lfloor d/2\rfloor +2$. Many explicit examples are given, along with techniques for producing others.

25 pages