特殊三元型分解的极小性和唯一性
Minimality and uniqueness for decompositions of specific ternary forms
摘要：
本文讨论了一类特殊的三值形式的秩的计算和可识别性。通常，人们知道给定形式的一些短的华林分解，问题是确定分解是否是最小的和唯一的。我们展示了如何分析的希尔伯特-伯奇矩阵的一组点代表的分解可以解决这个问题的情况下，三元形式。此外，当分解不是唯一的，我们展示了如何联络的过程可以提供替代，也许更短，分解。我们给出了一个明确的算法，测试我们的标准的情况下，三元形式的程度$9$的极小。这是第一个数值的情况下，出现了一个新的现象：跨度的$18$一般权力的线性型包含点（次属）秩$18$，但它也包含点的秩是$17$，由于存在第二个较短的分解，这是完全不同的给定的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
