On polynomial solutions to the minimal surface equation

We are interested in finding nonlinear polynomials P on Rⁿ that solves the minimal surface equation. We first prove a structure theorem on such polynomials. We show that the highest degree term Pₘ must factor as P₀ᵏQₘ where k is odd, Qₘ≥ 0 on Rⁿ and P₀ is irreducible, and that the level sets of Pₘ are all area minimizing. Moreover, the unique tangent cone of graph P at ∞ is ₀=0\. If k≥ 3, we know further that lower order terms down to some degree are divisible by P₀. We also show that P must contain terms of both high and low degree. In particular, it cannot be homogeneous. As a corollary, we prove that there is no quadratic or cubic polynomial solution. Using a general decay lower bound of Green's functions on minimal hypersurfaces we are able to show that deg P₀+k⁻¹deg Qₘ≤ n-2. Finally, we prove that a polynomial minimal graph cannot have C× Rˡ as its tangent cone at ∞ where l≥ 1 and C is any isoparametric cone. We also show that the existence of a nonlinear polynomial solutions on R⁸ will imply the existence of an area minimizing but not strictly minimizing cubic cone in R⁸. These results indicate that finding an explicit polynomial solution can be hard.