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

Abstract: Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ 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.