The Waring loci of ternary quartics

Generic Power Sum Decompositions and Bounds for the Waring Rank

A notion of open rank, related with generic power sum decompositions of forms, has recently been introduced in the literature. The main result here is that the maximum open rank for plane quartics is eight. In particular, this gives the first example of $n,d$, such that the maximum open rank for degree $d$ forms that essentially depend on $n$ variables is strictly greater than the maximum rank. On one hand, the result allows to improve the previously known bounds on open rank, but on the other hand indicates that such bounds are likely quite relaxed. Nevertheless, some of the preparatory resul…

函数域上 Waring 问题中超过二倍次数时的渐近性

The asymptotic in Waring's problem over function fields beyond twice the degree

本文证明函数域上 Waring 问题在 n>2d 时的预期渐近性，通过聚合小弧估计和相交理论获得最优范围及显式下界。

Every Ternary Quintic is a Sum of Ten Fifth Powers

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree $d$ (with coefficients in an algebraically closed field of characteristic zero) is known only for $d\le 4$. The best upper bound that is known for $d=5$ is twelve, and in this work we lower it to ten.

Inverting catalecticants of ternary quartics

We study the reciprocal variety to the linear space of symmetric matrices (LSSM) of catalecticant matrices associated with ternary quartics. With numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of the LSSM. We provide a geometric explanation to why equality between these two invariants is not reached, as opposed to the case of binary forms, by describing the intersection of the reciprocal variety and the orthogonal of the LSSM in the rank loci. Moreover, we prove that only the rank-$1$ locus, namely the Veronese surface $ν_4(\mathbb{P}^2)$, contributes to the degree o…

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

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 $\operatorname{r_{max}}(3,d)\le\left\lfloor\left(d^2+6d+1\right)/4\right\rfloor$.