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

Alessandro De Paris

Published 2015-10-27Version 1

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