Polynomials Nonnegative on Noncompact Subsets of the Plane Open Access

Abstract

Abstract
Polynomials Nonnegative on Noncompact Subsets of the Plane
By Ha Ngoc Nguyen

In 1991, Schmüdgen proved that if f is a polynomial in n variables with real coefficients such that f > 0 on a compact basic closed semialgebraic set K ⊆ Rⁿ, then there always exists an algebraic expression showing that f is positive on K. Then in 1999, Scheiderer showed that if K is not compact and its dimension is 3 or more, there is no analogue of Schmüdgen's Theorem. However, in the noncompact two-dimensional case, very little is known about when every f positive or nonnegative on a noncompact basic closed semialgebraic set K ⊆ R² has an algebraic expression proving that f is nonnegative on K. Recently, M. Marshall answered a long-standing question in real algebraic geometry by showing that if f ∈ R[x, y] and f ≥ 0 on the strip [0, 1] × R, then f has a representation f = σ0 + σ1x(1 − x), where σ0, σ1 ∈ R[x, y] are sums of squares.

This thesis gives some background to Marshall's result, which goes back to Hilbert's 17th problem, and our generalizations to other noncompact basic closed semialgebraic sets of R² which are contained in strip. We also give some negative results.

Polynomials Nonnegative on Noncompact Subsets of the Plane
By
Ha Ngoc Nguyen
M.S., Emory University, 2008
B.S., with Honors in Mathematics, University of California, Los Angeles, 2005
Advisor: Victoria Powers, Ph.D.
A dissertation submitted to the Faculty of the James T. Laney School of Graduate Studies of Emory University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics and Computer Science 2010

Table of Contents

1 Introduction 1

2 Preliminaries 4

2.1 Positivity and Sums of Squares . . . . . . . . . . . . . . . . . 4

2.2 Nonnegative Polynomials in R² . . . . . . . . . . . . . . . . . 11

3 Polynomials Nonnegative on Half-strips in the Plane 14

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Half-strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Further Examples in [0,1] × R⁺ . . . . . . . . . . . . . . . . . 21

4 Polynomials Nonnegative on Strips in the Plane 27

4.1 Reduction to a Positive Leading Coefficient . . . . . . . . . . . 28

4.2 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Representations of f by Analytic Functions . . . . . . . . . . . 35

4.4 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . 37

5 Conclusion and Future Work 42

Bibliography 44

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	real algebra positive polynomials and sums of squares
Committee Chair / Thesis Advisor	Powers, Victoria, Emory University
Committee Members	Raman, Parimala, Emory University Brussel, Eric, Emory University

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