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

TL;DR

This paper investigates the maximum Waring rank of ternary homogeneous polynomials of degree d, establishing its asymptotic behavior as approximately d^2/4 with an explicit upper bound.

Contribution

The paper determines the asymptotic leading term for the maximum rank of ternary forms, providing new bounds for degrees greater than 3.

Findings

Maximum rank for ternary forms grows as d^2/4 + O(d).

Established an explicit upper bound for the maximum rank.

Extended understanding of Waring ranks for specific polynomial degrees.

Abstract

Let $\operatorname{r_{max}}(n,d)$ be the maximum Waring rank for the set of all homogeneous polynomials of degree $d>0$ in $n$ indeterminates with coefficients in an algebraically closed field of characteristic zero. To our knowledge, when $n,d\ge 3$, the value of $\operatorname{r_{max}}(n,d)$ is known only for $(n,d)=(3,3),(3,4),(3,5),(4,3)$. We prove that $\operatorname{r_{max}}(3,d)=d^2/4+O(d)$ as a consequence of the upper bound $\operatorname{r_{max}}(3,d)\le\left\lfloor\left(d^2+6d+1\right)/4\right\rfloor$.