Index formulas for higher order Loewner vector fields

Let be the Cauchy–Riemann operator and f be a Cn real-valued function in a neighborhood of 0 in R2 in which for all z≠0. In such cases, is known as a Loewner vector field due to its connection with Loewner's conjecture that the index of such a vector field is bounded above by n. The n=2 case of Loewner's conjecture implies Carathéodory's conjecture that any C2-immersion of S2 into R3 must have at least two umbilics. Recent work of F. Xavier produced a formula for computing the index of Loewner vector fields when n=2 using data about the Hessian of f. In this paper, we extend this result and establish an index formula for for all n⩾2. Structurally, our index formula provides a defect term, which contains geometric data extracted from Hessian-like objects associated with higher order derivatives of f.