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arXiv 1510.08048math.AGmath.AC
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给定次数的三元型的最大秩的渐近首项

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

Alessandro De Paris

备注：
v1: 10 pages. v2: extended introduction and some mistakes corrected
发表信息：
Final version in Linear Algebra Appl. 500 (2016) 15--29
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摘要：

令$\operatorname{r_{max}}(n,d)$是$n$中所有$d>0$次齐次多项式的集合的最大Waring秩，其中系数在特征为零的代数闭域中。据我们所知，当$n,d\ge 3$时，$\operatorname{r_{max}}(n,d)$的值仅对于$(n,d)=(3,3),(3,4),(3,5),(4,3)$是已知的。我们证明了$\operatorname{r_{max}}(3,d)=d^2/4+O(d)$作为上界$\operatorname{r_{max}}(3,d)\le\left\lfloor\left(d^2+6d+1\right)/4\right\rfloor$的结果。

英文摘要：

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

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