Geometry of Hermitian Algebraic Functions. Quotients of Squared Norms

2005/04/11 by Dror Varolin, Varolin, Dror
Mathematics · #14E15 #32L99 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.CV #msc:14E15 #msc:32L99

paper · pdf · doi:10.48550/arxiv.math/0504229

Major revision. Some errors in the proof corrected, and other incorrect results removed, but the main result remains the same

openalex publication_date 2005/04/11 · arxiv created 2006/07/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Hermitian algebraic functions were introduced by Catlin and D'Angelo under the name of "globalizable metrics". Catlin and D'Angelo proved that any Hermitian algebraic function without non-trivial zeros is a quotient of squared norms, thus giving an answer to a Hermitian analogue of Hilbert's 17th problem in the non-degenerate case. The result was independently proved somewhat earlier by D. Quillen in a special case, and using different methods. In this paper, we characterize all Hermitian algebraic functions that are quotients of squared norms.

Citations

• Smoothness, Semistability, and Toroidal Geometry
• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

Related

• Hermitian analogues of Hilbert's 17-th problem
• The Hermitian Distance degree of an Algebraic Variety
• Eventual positivity of Hermitian algebraic functions and associated integral operators
• Algebraic properties of Hermitian sums of squares, II
• Binary normal regular Hermitian lattices over imaginary quadratic fields