论文标题
关于一般多项式的强度
On the strength of general polynomials
论文作者
论文摘要
切片分解是均匀多项式作为线性因子的形式总和的表达。强度分解是均质多项式作为可还原形式的总和的表达。多项式的切片等级和强度分别是此类分解的最小长度。切片等级是强度的上限，这两个值之间的差距可能是任意的。但是，与Catalisano等人的猜想一致。在可简化形式的割线品种的尺寸上，我们猜想平等对一般形式的构成。通过使用弗弗格（Fröberg）在希尔伯特（Hilbert）系列的一系列理想中使用较弱版本的猜想，我们表明我们的猜想最高为$ 7 $，程度为$ 9 $。
A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fröberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree $7$ and in degree $9$.
