Foundations of Computational Mathematics

, Volume 17, Issue 2, pp 423–465

# Generating Polynomials and Symmetric Tensor Decompositions

• Jiawang Nie
Article

## Abstract

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.

## Keywords

Symmetric tensor Tensor rank Generating polynomial Generating matrix Symmetric tensor decomposition Polynomial system

15A69 65F99

## Notes

### Acknowledgments

The author would like to thank Lek-Heng Lim, Luke Oeding, and two anonymous referees for the useful comments. The research was partially supported by the NSF Grants DMS-0844775 and DMS-1417985.

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