Non-polynomial entire solutions to $\sigma_{k}$ equations

TL;DR

This paper constructs explicit non-polynomial solutions to the Hessian equation _k(D^2 u)=1 in _n for cases where 2k  n+1, expanding the known solution space for these fully nonlinear PDEs.

Contribution

It provides the first known explicit non-polynomial entire solutions to _k equations for the specified parameter range, broadening understanding of solution types.

Findings

Existence of non-polynomial solutions on _n for 2k  n+1

Solutions are explicit and valid on all of _n

Extends the class of known solutions to Hessian equations

Abstract

For $2k \leq n+1$, we exhibit non-polynomial solutions to the Hessian equation \[ \sigma_{k}(D^{2}u)=1 \] on all of $\mathbb{R}^{n}.$