Differential characterization of quadratic surfaces
Bartłomiej Zawalski
TL;DR
The paper characterizes graphs of functions whose graphs lie on quadratic surfaces in terms of a simple, explicit system of third-order PDEs. It develops a differential-algebraic framework and shows that the associated annihilator is generated by two third-order operators forming a reduced Gröbner basis, linking polynomial PDEs to generalized Wronskians. A smoothing analysis demonstrates that generic weak solutions are actually infinitely differentiable and connect to holomorphicity via a constructed pair of functions satisfying Cauchy–Riemann equations. When the Hessian determinant is positive somewhere, the graph must lie on a quadratic surface; if the determinant is non-positive, the domain decomposes, with pieces corresponding to developable, Catalan, or doubly-ruled surfaces. This provides a 2D analogue of classical affine-geometric characterizations and has potential applications in convex geometry and related areas.
Abstract
Let $f\in W^{3,1}_{\mathrm{loc}}(Ω)$ be a function defined on a connected open subset $Ω\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of third-order partial differential equations unless the Hessian determinant of $f$ is non-positive everywhere on $Ω$. Moreover, we will prove that the system is, in a sense, the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somewhat related to holomorphic functions.