On the number of Waring decompositions for a generic polynomial vector

Abstract

Abstract. We prove that a general polynomial vector (f 1 , f 2 , f 3 ) in three homogeneous variables of degrees (3, 3, 4) has a unique Waring decomposition of rank 7. This is the first new case we are aware, and likely the last one, after five examples known since 19th century and the binary case. We prove that there are no identifiable cases among pairs (f 1 , f 2 ) in three homogeneous variables of degree (a, a + 1), unless a = 2, and we give a lower bound on the number of decompositions. The new example was discovered with Numerical Algebraic Geometry, while its proof needs Nonabelian Apolarity.

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

“…, forms of multidegree (2, 2, 2, 2) in four sets of three variables; here, the generic partially-symmetric-rank is four (this is a classical result; see also [204]); • S 2 C 3 ⊗S 3 C 3 , i.e., forms of multidegree (2,3) in two sets of three variables; here, the generic partiallysymmetric-rank is four [207]; • S 2 C 3 ⊗ S 2 C 3 ⊗ S 4 C 3 , i.e., forms of multidegree (2,2,4) in three sets of three variables; here, the generic partially-symmetric-rank is seven [204].…”

Decomposability of Tensors

“…When we come to partially-symmetric tensors (which are related to Segre-Veronese varieties, as we described in Section 4.2), a complete classification of generically-identifiable cases is not known, but it is known that it happens in the following cases; see [203]. S 2 C n+1 ⊗ S 2 C n+1 , i.e., forms of multidegree (2, 2) in two sets of n + 1 variables; here, the generic partially-symmetric-rank is n + 1 [205]; • S 2 C 3 ⊗ S 2 C 3 ⊗ S 2 C 3 ⊗ S 2 C 3 , i.e., forms of multidegree (2, 2, 2, 2) in four sets of three variables; here, the generic partially-symmetric-rank is four (this is a classical result; see also [203]); • S 2 C 3 ⊗ S 3 C 3 , i.e., forms of multidegree (2, 3) in two sets of three variables; here, the generic partially-symmetric-rank is four [206]; • S 2 C 3 ⊗ S 2 C 3 ⊗ S 4 C 3 , i.e., forms of multidegree (2, 2, 4) in three sets of three variables; here, the generic partially-symmetric-rank is seven [203].…”

Generic identifiability of pairs of ternary forms

“…They work in a slightly different language and phrase their statement in terms of geometric properties of a rational normal scroll. Their result applies to tuples of binary forms, not just pairs, see also [AGMO18,Theorem 1.3]. Roughly speaking, the general pair of binary forms is identifiable, as long as the polynomial system (2) is square and c is not too small compared to d.…”