# High-rank ternary forms of even degree

## TL;DR

This paper constructs high-rank ternary forms of even degree, establishing new lower bounds for the maximum rank of degree d forms in three variables over algebraically closed fields.

## Contribution

It provides explicit examples of high-rank ternary forms for even degrees, extending previous results and establishing a new lower bound for maximum rank.

## Key findings

- Constructed ternary forms with rank exceeding monomials
- Established lower bound for maximum rank in three variables
- Extended results to even degrees in the rank study

## 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 $\operatorname{r_{max}}(n,d)$ denotes the maximum rank of degree $d$ forms in $n$ variables with coefficients in an algebraically closed field of characteristic zero.