We prove that a general polynomial vector (f1,f2,f3) 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 (f1,f2) 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.
On the number of Waring decompositions for a generic polynomial vector / Angelini, Elena; Galuppi, Francesco; Mella, Massimiliano; Ottaviani, Giorgio. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - STAMPA. - 222:(2018), pp. 950-965. [10.1016/j.jpaa.2017.05.016]
On the number of Waring decompositions for a generic polynomial vector
Ottaviani, GiorgioMembro del Collaboration Group
2018
Abstract
We prove that a general polynomial vector (f1,f2,f3) 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 (f1,f2) 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.File | Dimensione | Formato | |
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