Linearly dependent powers of binary quadratic forms

Abstract

Given an integer d ≥ 2, what is the least r so that there is a set of binary quadratic forms {f 1 , . . . , f r } for which {f d j } is non-trivially linearly dependent? We show that if r ≤ 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.

Equal sums of two cubes of binary quadratic forms

“…We need an old fact about simultaneous diagonalization; there doesn't seem to be a standard easy-to-find modern proof, a different proof is shown in [15,Thm.3.2].…”

“…which, as in [15], we call the tame case; otherwise we are in the wild case. If a = 0, then it follows from (3.1) that y divides f 3 and f 4 , and by Lemma 2.4, this cannot happen, so a = 0.…”

“…The author foolishly believed that an algebraic approach would easily lead to all solutions, and posted a proof-free online set of notes [13] in 2000. Eventually, it has produced this article and an earlier companion paper studying higher powers, [15]. He wishes to thank his present and former colleagues Michael Bennett, Bruce Berndt, Nigel Boston, Dan Grayson and Jeremy Rouse for helpful conversations, and Andrew Bremner, Noam Elkies and Michael Hirschhorn for encouraging and useful emails over the years.…”