论文标题
特定三元形式分解的最小和独特性
Minimality and uniqueness for decompositions of specific ternary forms
论文作者
论文摘要
本文涉及等级的计算和特定三元形式的可识别性。通常，人们知道给定形式的一些短暂的警告分解，问题是确定分解是否最小和独特。我们展示了代表分解点的一组点的希尔伯特 - 比奇矩阵的分析可以在三元形式的情况下解决此问题。此外，当分解不是唯一的时候，我们展示了联络过程如何提供替代（可能更短的分解）。我们提供了一种明确的算法，该算法测试了我们对三元形式的$ 9 $的最小化标准。这是出现新的phaenomenon的第一个数值案例：线性形式的$ 18 $的一般力量的跨度包含（亚基）等级$ 18 $的点，但由于存在第二个短分解，其等级为$ 17 $，这与给定的一个完全不同。
The paper deals with the computation of the rank and the identifiability of a specific ternary form. Often, one knows some short Waring decomposition of a given form, and the problem is to determine whether the decomposition is minimal and unique. We show how the analysis of the Hilbert-Burch matrix of the set of points representing the decomposition can solve this problem in the case of ternary forms. Moreover, when the decomposition is not unique, we show how the procedure of liaison can provide alternative, maybe shorter, decompositions. We give an explicit algorithm that tests our criterion of minimality for the case of ternary forms of degree $9$. This is the first numerical case in which a new phaenomenon appears: the span of $18$ general powers of linear forms contains points of (subgeneric) rank $18$, but it also contains points whose rank is $17$, due to the existence of a second shorter decomposition which is completely different from the given one.
