Counterexamples to a Conjecture of Fröberg, Lundqvist, Oneto and Shapiro on Maximal k-Ranks of Binary Forms

For a complex binary form \(f\) of degree \(kd\), the \(k\)-rank of \(f\) is the least number of \(k\)-th powers of binary forms of degree \(d\) whose sum is \(f\). Fröberg, Lundqvist, Oneto and Shapiro conjectured (Arnold Math. J. 4 (2018), Conjecture 1.9) that for all \(k\) and \(d\) the maximal \(k\)-rank of binary forms of degree \(kd\) equals \(k\), and that the forms \(\ell_1\ell_2^{kd-1}\), with non-proportional linear forms \(\ell_1,\ell_2\), attain it. We show that \(xy^{kd-1}\) is a sum of fewer than \(k\) \(k\)-th powers of forms of degree \(d\) for every \(d\ge2\) when \(k\in\{5,6,7,8,9\}\) or \(k\) is odd with \(7\le k\le201\); so the conjecture is false for all these \((k,d)\). For \(k=7\) the counterexample is a closed-form identity whose proof needs no computer: \(xy^{13}\) is a linear combination of six \(7\)th powers of explicit binary quadratic forms, built from \(\sqrt{165}\), a cube root, two square roots and a cube root of unity, and hence a sum of six \(7\)th powers. The same construction gives \(k-1\) terms for odd \(k\ge7\) whenever three conditions on two explicit integer polynomials hold; they were checked by exact computer calculation for \(7\le k\le201\). This construction is a variant of Reznick's “synching” of powers of quadratic forms. For \(k=5,6,8,9\) the identities, with \(4,5,6,7\) terms, are computer-assisted: an interval Newton test of Krawczyk type, carried out in exact rational and interval arithmetic by several independent programs, proves that an exact solution exists next to an approximate one. By results of Lundqvist, Oneto, Reznick and Shapiro, of Carlini and Oneto, and of Bik, Draisma, Oneto and Ventura, the conjecture holds at \((k,d)=(3,2),(3,3),(3,4),(4,2)\), so \((5,2)\) is the first failure in the order of \(k\) for \(d=2\); the case \(k=4\), \(d\ge3\) remains open, and so does the first clause alone (maximal \(k\)-rank equal to \(k\)) outside the known cases. The same identities answer Problem D of the source, on \(k\)-ranks of binary monomials, in the negative, and they give upper bounds below \(N\) in Heilbronn's problem on sums of \(N\)-th powers of entire functions and in its analogue for polynomials, the Waring number of \(\mathbb{C}[t]\). We found no such bound in the literature accessible to us, but two primary sources could not be read. This is an unrefereed note.