TY - JOUR

T1 - Linearly dependent powers of binary quadratic forms

AU - Reznick, Bruce

N1 - Funding Information:
The author was supported by Simons Collaboration Grant 280987. MSC2010: primary 11E76, 11P05, 14M99; secondary 11D25, 11D41. Keywords: polynomial identities, super-Fermat problem for forms.
Publisher Copyright:
© 2019 Mathematical Sciences Publishers.

PY - 2019

Y1 - 2019

N2 - Given an integer d ≥ 2, what is the smallest r so that there is a set of binary quadratic formsg (f1,...,fr) for which (fdj) is nontrivially linearly dependent? We show that ifr ≤4, then d ≤5, and for d ≥ 4, construct such a set with r =d/2+2. Many explicit examples are given, along with techniques for producing others.

AB - Given an integer d ≥ 2, what is the smallest r so that there is a set of binary quadratic formsg (f1,...,fr) for which (fdj) is nontrivially linearly dependent? We show that ifr ≤4, then d ≤5, and for d ≥ 4, construct such a set with r =d/2+2. Many explicit examples are given, along with techniques for producing others.

KW - Polynomial identities

KW - Super-Fermat problem for forms

UR - http://www.scopus.com/inward/record.url?scp=85079269570&partnerID=8YFLogxK

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U2 - 10.2140/pjm.2019.303.729

DO - 10.2140/pjm.2019.303.729

M3 - Article

AN - SCOPUS:85079269570

VL - 303

SP - 729

EP - 755

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -