High-rank ternary forms of even degree

Abstract

We exhibit, for each 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 the lower bound \[\operatorname{r_{max}}(3,d)\ge\left\lfloor\frac{d^2+2d+5}4\right\rfloor\] for $d\ge 2$, where ${\mathrm{r}}_{\max }(n,d)$ denotes the maximum rank of degree $d$ forms in $n$ variables with coefficients in an algebraically closed field of characteristic zero.

References in corpus (1)

• The asymptotic leading term for maximum rank of ternary forms of a given degree