Linearly dependent powers of binary quadratic forms

Given an integer $d \geq 2$ , what is the smallest $r$ so that there is a set of binary quadratic forms ${f_{1}, \dots, f_{r}}$ for which ${f_{j}^{d}}$ is nontrivially linearly dependent? We show that if $r \leq 4$ , then $d \leq 5$ , and for $d \geq 4$ , 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

Vol. 303, No. 2, 2019

Bruce Reznick

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors