High-rank ternary forms of even degree

Abstract

We exhibit, for each positive even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives a lower bound of

$$\begin{aligned} \left\lfloor \frac{d^2+2d+5}{4}\right\rfloor \;, \end{aligned}$$

for $d\ge 2$, on the maximum rank of degree d ternary forms with coefficients in an algebraically closed field of characteristic zero.