Linearly dependent powers of binary quadratic forms

Bruce Reznick

Vol. 303 (2019), No. 2, 729–755

DOI: 10.2140/pjm.2019.303.729

Abstract

Given an integer $d\ge 2$, what is the smallest $r$ so that there is a set of binary quadratic forms $\left\{f_1,\dots ,f_r\right\}$ for which $\left\{f_j^d\right\}$ is nontrivially 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.

Keywords

polynomial identities, super-Fermat problem for forms

Mathematical Subject Classification 2010

Primary: 11E76, 11P05, 14M99

Secondary: 11D25, 11D41

Milestones

Received: 18 April 2019

Revised: 27 June 2019

Accepted: 27 June 2019

Published: 4 January 2020