Generic identifiability of pairs of ternary forms

On the identifiability of ternary forms

We describe a new method to determine the minimality and identifiability of a Waring decomposition $A$ of a specific form (symmetric tensor) $T$ in three variables. Our method, which is based on the Hilbert function of $A$, can distinguish between forms in the span of the Veronese image of $A$, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length $r$ of the decomposition, from $2$ up to the generic rank, a range which was not achievable before. Though …

关于同时 Waring 可识别性猜想的一个反例

A counterexample to a conjecture on simultaneous Waring identifiability

本文通过计算机辅助程序构造反例，证明在 $d=4$ 时多项式向量的同时 Waring 分解不唯一，推翻了关于同时 Waring 可识别性完全分类的猜想。

On a geometric method for the identifiability of forms

We introduce a new criterion which tests if a given decomposition of a given ternary form $T$ of even degree is unique. The criterion is based on the analysis of the Hilbert function of the projective set of points $Z$ associated to the decomposition, and on the Terracini's Lemma which describes tangent spaces to secant varieties. The criterion works in a range for the length of the decomposition which is equivalent to the range in which the reshaped Kruskal's criterion (see [1]) works. Our criterion determines an algorithm for the identifiability of $T$ which is sensibly faster than algorithm…

Minimality and uniqueness for decompositions of specific ternary forms

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 criterio…

任意基上四次代数的参数化

Parametrizing quartic algebras over an arbitrary base

我们参数化四次交换代数在任何基环或计划（等价有限，平坦度四$S$ -计划），其三次预解式，对三元二次型的基础。这推广了Bhargava的参数化的四次环与他们的三次预解环在$\mathbb{Z}$通过对整数三元二次型，以及Casnati和Ekedahl的建设Gorenstein四次覆盖的某些秩2家庭的三元二次型。本文给出了由任意一对三元二次型构成的四次代数的一个尺规作图法，证明了这个构造与基变交换，并且与Bhargava在$\mathbb{Z}$上的显式构造一致.

双线性反问题的最优内射性条件及其在反褶积问题可识别性中的应用

Optimal Injectivity Conditions for Bilinear Inverse Problems with Applications to Identifiability of Deconvolution Problems

研究了稀疏性和子空间约束下双线性反问题的可辨识性。我们表明，到一个全球尺度模糊，几乎所有这样的地图是内射的一组对稀疏向量的测量$m$超过$2(s_1+s_2)-2$，其中$s_1$和$s_2$表示的稀疏性的两个输入向量，和内射的一组对向量躺在已知的子空间的尺寸$n_1$和$n_2$，如果$m\geq 2(n_1+n_2)-4$。我们还证明了这两个界限是紧的，在这个意义上说，一个不能有内射性为一个较小的数量的测量。我们的证明技术来自代数几何。作为应用，我们导出了反卷积问题的最优可识别性条件，从而改进了Li等人[1]的最近工作。