### Abstract

Let R [X] : = R [X_{1}, ..., X_{n}]. Pólya's Theorem says that if a form (homogeneous polynomial) p ∈ R [X] is positive on the standard n-simplex Δ_{n}, then for sufficiently large N all the coefficients of (X_{1} + ⋯ + X_{n})^{N} p are positive. The work in this paper is part of an ongoing project aiming to explain when Pólya's Theorem holds for forms if the condition "positive on Δ_{n}" is relaxed to "nonnegative on Δ_{n}", and to give bounds on N. Schweighofer gave a condition which implies the conclusion of Pólya's Theorem for polynomials f ∈ R [X]. We give a quantitative version of this result and use it to settle the case where a form p ∈ R [X] is positive on Δ_{n}, apart from possibly having zeros at the corners of the simplex.

Original language | English (US) |
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Pages (from-to) | 1285-1290 |

Number of pages | 6 |

Journal | Journal of Symbolic Computation |

Volume | 44 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2009 |

### Keywords

- Positive polynomials
- Pólya's Theorem
- Sums of squares

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics

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## Cite this

*Journal of Symbolic Computation*,

*44*(9), 1285-1290. https://doi.org/10.1016/j.jsc.2008.04.021