关于一般多项式的强度
On the strength of general polynomials
摘要：
切片分解是将齐次多项式表示为具有线性因子的形式之和。强度分解是将齐次多项式表示为可约形式之和。多项式的切片秩和强度分别是这种分解的最小长度。切片秩是强度的上界，并且这两个值之间的差距可以是任意大的。然而，在符合Catalisano等人的猜想。对尺寸割线品种品种的品种可约形式，我们猜想，平等举行的一般形式。利用Fröberg猜想的一个较弱的版本，证明了我们的猜想在$7$阶和在$9$阶都成立.
英文摘要：
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 $9$.
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