Polynomials nonnegative on the cylinder

2016/07/29 by Claus Scheiderer, Scheiderer, Claus, Sebastian Wenzel +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1607.08705

openalex publication_date 2016/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 2010, Marshall settled the strip conjecture, according to which every polynomial in ℝ[x,y], nonnegative on the strip [-1,1]×ℝ, is a sum of squares and of squares times 1-x². We consider affine nonsingular curves C over ℝ with C(ℝ) compact, and study the question whether every f in ℝ[C][y], nonnegative on C(ℝ)×ℝ, is a sum of squares in ℝ[C][y]. We give an affirmative answer under the condition that f has only finitely many zeros in C(ℝ)×ℝ. For C the circle x₁²+x₂²=1, we prove the result unconditionally.

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