### Abstract

A famous theorem of Hilbert from 1888 states that for given n and d, every positive semidefinite (psd) real form of degree 2d in n variables is a sum of squares (sos) of real forms if and only if n=2 or d=1 or (n,2d)=(3,4). In 1976, Choi and Lam proved the analogue of Hilbert's Theorem for symmetric forms by assuming the existence of psd not sos symmetric n-ary quartics for n≥5. In this paper we complete their proof by constructing explicit psd not sos symmetric n-ary quartics for n≥5.

Original language | English (US) |
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Pages (from-to) | 114-120 |

Number of pages | 7 |

Journal | Linear Algebra and Its Applications |

Volume | 496 |

DOIs | |

State | Published - May 1 2016 |

### Keywords

- Positive polynomials
- Sums of squares
- Symmetric forms

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

Goel, C., Kuhlmann, S., & Reznick, B. (2016). On the Choi-Lam analogue of Hilbert's 1888 theorem for symmetric forms.

*Linear Algebra and Its Applications*,*496*, 114-120. https://doi.org/10.1016/j.laa.2016.01.024