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

Abstract

Let ${\mathrm{r}}_{\max }(n,d)$ 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 ${\mathrm{r}}_{\max }(n,d)$ is known only for $(n,d)=(3,3),(3,4),(3,5),(4,3)$. We prove that ${\mathrm{r}}_{\max }(3,d)=d^2/4+O(d)$ as a consequence of the upper bound ${\mathrm{r}}_{\max }(3,d)\le \lfloor \left(d^2+6d+1\right)/4\rfloor$.

v1: 10 pages. v2: extended introduction and some mistakes corrected

References in corpus (2)

• Geometric lower bounds for generalized ranks
• Every Ternary Quintic is a Sum of Ten Fifth Powers

Cited by in corpus (3)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• Generic Power Sum Decompositions and Bounds for the Waring Rank
• High-rank ternary forms of even degree