IJPAM: Volume 93, No. 6 (2014)

ON THE UNIQUENESS FOR THE SYMMETRIC TENSOR
RANK OF TRIVARIATE POLYNOMIALS; A LOCAL
UNIQUENESS FOR MULTIVARIATE POLYNOMIALS

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY


Abstract. For all integers $m\ge 2$ and $d\ge 3$ and $x>0$. Let $\nu _d: \mathbb {P}^m\to \mathbb {P}^N$, $N:= \binom{m+d}{m}-1$, be the Veronese embedding. We discuss the uniqueness (only for trivariate polynomials) and the local uniqueness of a decomposition of a polynomial into powers of linear forms in the following sense. Take $P\in \mathbb {P}^N$. Let $S(m,x,d,P)$ be the set of all $S\subset \mathbb {P}^m$ such that $\sharp (S)=x$, $P\in \langle \nu _d(S)\rangle$ (where $\langle \ \rangle$ is the linear span), and $P\notin \langle \nu _d(S')\rangle$ for any $S'\subsetneq S$. We prove that $S(m,x,d,P) =\{S\}$ (resp. $S$ is a isolated point of $S(m,x,d,P)$) if $m=2$, $x < (d^2+3d)/8$ and $S$ has the general uniform position (resp. $\sharp (S) \le \binom{m+\lfloor (d-1)/2\rfloor}{m}$ and $S$ has general postulation). We do the same for zero-dimensional schemes (scheme rank or cactus rank).

Received: March 1, 2014

AMS Subject Classification: 14N05

Key Words and Phrases: symmetric tensor rank, trivariate polynomial, zero-dimensional scheme, multivariate polynomial

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DOI: 10.12732/ijpam.v93i6.7 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 93
Issue: 6
Pages: 807 - 812

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).