On the number of Waring decompositions for a generic polynomial vector

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.

arXiv comment17 pages, a Macaulay2 code as ancillary file

References in corpus (6)

• Eigenvectors of tensors and algorithms for Waring decomposition
• Birational aspects of the Geometry of Varieties of Sum of Powers
• Decomposing Tensors into Frames
• Grassmann secants, identifiability, and linear systems of tensors
• Homotopy techniques for tensor decomposition and perfect identifiability
• Grassmann defectivity à la Terracini

Cited by in corpus (7)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• Partially symmetric variants of Comon's problem via simultaneous rank
• Waring decompositions and identifiability via Bertini and Macaulay2 software
• Projective and birational geometry of Grassmannians and other special varieties
• On complex and real identifiability of tensors
• Generic identifiability of pairs of ternary forms
• Partially complex ranks for real projective varieties