2023/09/21 by Giulio Bresciani, Bresciani, Giulio
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2309.12192
openalex publication_date 2023/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that a smooth, complex plane curve C of odd degree can be defined by a polynomial with real coefficients if and only if C is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More generally, we prove that a plane curve C over an algebraically closed field K of characteristic 0 with field of moduli kC⊂ K is defined by a polynomial with coefficients in k', where k'/kC is an extension with [k':kC]≤ 3 and [k':kC]| deg C.