Decomposition algorithms for tensors and polynomials

TL;DR

This paper introduces algorithms for tensor and polynomial decompositions, providing methods to compute and verify the uniqueness of such decompositions, with specific applications to plane curves and explicit Magma implementations.

Contribution

It presents new algorithms for decomposing polynomials and tensors, including specific cases like plane quintics and sextics, and offers Magma code implementations.

Findings

Algorithms successfully decompose specific plane curves.

Unique decompositions are established for certain tensors.

Magma implementations are provided for practical use.

Abstract

We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the decomposition of a general plane quintic in seven powers, and of a general space cubic in five powers; the two decompositions of a general plane sextic of rank nine, and the five decompositions of a general plane septic. Furthermore, we give Magma implementations of all our algorithms.