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 2, what is the smallest r so that there is a set of binary quadratic forms {f1,,fr} for which {fjd} is nontrivially linearly dependent? We show that if r 4, then d 5, and for d 4, construct such a set with r = d2 + 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