#### Vol. 303, No. 2, 2019

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Linearly dependent powers of binary quadratic forms

### Bruce Reznick

Vol. 303 (2019), No. 2, 729–755
##### 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 $r=⌊d∕2⌋+2$. Many explicit examples are given, along with techniques for producing others.

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##### 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
##### Authors
 Bruce Reznick Department of Mathematics University of Illinois at Urbana–Champaign Urbana, IL United States