On the identifiability of ternary forms

Abstract

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 the method in principle can handle all cases of specific ternary forms, we introduce and describe it in details for forms of degree 8.

Minimality and uniqueness for decompositions of specific ternary forms

“…Papers [10], [8], [9] are based on the study of Hilbert functions of points. What is really new in the present paper, as well as in [4], is the observation that given a finite set A ⊂ P 2 , the possible alternative (maybe even shorter) decompositions of any ternary form F in the span of ν d (A) can be recovered geometrically, by playing with the Hilbert-Burch matrix of A and liaison. Thus, we can characterize the (algebraic) subset Θ of the span of ν d (A) consisting of forms for which A is non-redundant, but yet they have an alternative decomposition of length r ′ < r. Moreover, we produce concrete algorithms which guarantee that a form F does not lie in the 'bad' set Θ.…”

“…In the case of ternary forms, the reshaped Kruskal's criterion has been recently extended in [4], [7], [31].…”

“…On the contrary, for subgeneric values r < r g , uniqueness holds for a minimal Waring expression of a sufficiently general form F , except for a short list of cases (see [15]). On the other hand, for a specific decomposition of a specific form F , whose rank is unknown, all the known methods can guarantee its uniqueness or minimality only when the length r is much smaller than r g (see Section 3 of [4], for a concrete bound).…”

“…Our analysis follows the guidelines introduced in [4]. We attack the problem with a set of tools typical for the study of the geometry of finite sets in projective spaces: Hilbert-Burch matrices, the Cayley-Bacharach property, and liaison.…”

On a geometric method for the identifiability of forms

“…, P 12 (this can be obtained by a direct computation on one specific point T , see e.g. [22] Claim 4.4). Thus a general T in the span of P 1 , .…”