On polynomial solutions to the minimal surface equation

We are interested in finding a nonlinear polynomial $P$ on $\mathbb{R}^n$ that solves the minimal surface equation. Even though no explicit solution is found in this article, we investigate constraints that a polynomial solution must obey. We first prove a structure theorem on such polynomials. We show that the highest degree term $P_m$ must factor as $p^kQ_m$ where $k$ is odd, $p$ is irreducible, and $Q_m\ge 0$ on $\mathbb{R}^n$ with $\{Q_m=0\}\subset\{p=0\}\cap\{\nabla p=0\}$. Moreover, the level sets of $P_m$ are all area-minimizing and the unique tangent cone of $\operatorname{graph} P$ at infinity is $\{p=0\}\times\mathbb{R}$. If $k\ge 3$, we know further that lower order terms down to some degree are divisible by $p$. We also show that $P$ must contain terms of both high and low degree. In particular, it cannot be homogeneous. As a consequence of the structure theorem, we get degree estimates for polynomial solutions. We have $\operatorname{deg} P\ge 4$ by ruling out cubic polynomial solutions. Using an extended eigenvalue estimate on the Jacobi operator by Zhu \cite{zhu2018first}, we are able to show that $\mu_n^-< \operatorname{deg} p +k^{-1}\operatorname{deg} Q_m< \mu_n^+$ where $\mu_n^\pm=\frac{n-1\pm\sqrt{(n-3)^2-4(n-2)}}{2}$. Finally, we prove that $\{p=0\}$ cannot be an isoparametric minimal cone. We also show that for a nonlinear polynomial solution on $\mathbb{R}^8$, we have $\operatorname{deg} p=3$ and that $\{p=0\}$ is an area-minimizing but not strictly minimizing cone in $\mathbb{R}^8$. These results give strong restrictions on possible polynomial solutions to the minimal surface equation.