Affine Homogeneous Surfaces with Hessian rank 2 and Algebras of Differential Invariants

Abstract

Consider a graphed holomorphic surface $u=F(x,y)$ in ${\mathbb{C}}_{x,y,u}^3$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential vector fields. Inspired by Olver's recurrence formulas, we study the algebra of $A(3)$ differential invariants of surfaces. We obtain necessary conditions for homogeneity of algebraic nature. Solving these conditions, we organise homogeneous models in inequivalent branches.

This work was supported in part by the Polish National Science Centre (NCN) via the grant number 2018/29/B/ST1/02583. 20 Pages, 3 diagrams, explicit expressions of relative invariants