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Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

Colin Tan, Wing-Keung To

Abstract

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set can be represented by some polynomial with only positive coefficients. This result generalizes a result of Pólya which corresponds to the case when the semi-algebraic set is the standard simplex. Our proof uses the Archimedean Representation Theorem from real algebra.

Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

Abstract

We consider polynomials on the intersection of the closed positive orthant with the height- level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set can be represented by some polynomial with only positive coefficients. This result generalizes a result of Pólya which corresponds to the case when the semi-algebraic set is the standard simplex. Our proof uses the Archimedean Representation Theorem from real algebra.
Paper Structure (5 sections, 6 theorems, 6 equations)

This paper contains 5 sections, 6 theorems, 6 equations.

Table of Contents

  1. Introduction and Statement of Result
  2. Main Result.
  3. The Archimedean Representation Theorem
  4. Proof of Theorem \ref{['thm: mainTheorem']}
  5. Acknowledgements.

Key Result

Theorem 1

Let $p\in{\mathbb{R}}[x_1, \dots, x_n]$ be a homogeneous polynomial such that $p > 0$ on $\Delta_{n}$. Then there exists some nonnegative integer $N_o$ such that for all integers $N \geq N_o$, all monomial terms of degree $N + \deg(p)$ in $(x_1 + \cdots + x_n)^N p$ have strictly positive coefficient

Theorems & Definitions (12)

  • Theorem 1: Pólya Polya1928
  • Theorem 2
  • Remark 3
  • Remark 4
  • Theorem 5: Archimedean Representation Theorem PrestelDelzell2001, Scheiderer2009 or Scheiderer2024
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Corollary 8
  • ...and 2 more