Planar vector field versions of Carathéodory's and Loewner's conjectures (Q1365661)

After summing up definitions and facts about Carathéodory's conjecture (all ovaloids must have at least two umbilics) and Loewner's conjecture (an ovaloid has no umbilic of index \(>1\)) [\textit{C. Gutierrez, F. Mercuri} and \textit{F. Sánchez-Bringas}, Exp. Math. 5, 33-37 (1996; Zbl 0862.57023)], the authors give a proof of the following result: Carathéodory's conjecture for \(C^r\) ovaloids (\(r=3,4,\dots,\infty\)) is equivalent to the following conjecture: if a planar \(C^r\) vector field defined in a neighborhood of the disk \(D(0,\rho)\) has, in a neighborhood of the circle \(\partial D(0,\rho)\), the form \((\beta_{xx} - \beta_{yy}, 2 \beta_{x,y})\) with \(\beta(x,y) = (a x^2 + b y^2)/(x^2 + y^2)\), then this vector field has at least two singularities in \(D(0,\rho)\).