Polynomial solutions of the minimal surface equation

Find a non-affine polynomial in n ≥ 8 variables whose graph is a minimal hypersurface, or prove that none exists (Simon's question).

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1	# Polynomial solutions of the minimal surface equation
2
3	Bernstein's theorem says that a minimal surface which is the graph of a function over the whole plane must itself be a plane. The same holds for graphs over $\mathbb{R}^n$ when $n \le 7$ and fails when $n \ge 8$, but all known counterexamples are transcendental. Leon Simon asked whether a polynomial can do it: the task is to find one, or to show there is none.
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5	## Problem
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7	A function $u\colon \mathbb{R}^n \to \mathbb{R}$ has a minimal graph in $\mathbb{R}^{n+1}$ exactly when it solves the minimal surface equation
8	$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1 + \|\nabla u\|^2}}\right) = 0.$$
9	Clearing the positive denominator, for a polynomial $u$ this is the polynomial identity
10	$$\bigl(1 + \|\nabla u\|^2\bigr)\,\Delta u \;-\; \sum_{i,j=1}^{n} u_i\, u_j\, u_{ij} \;=\; 0,$$
11	where $u_i = \partial u/\partial x_i$ and $u_{ij} = \partial^2 u/\partial x_i \partial x_j$.
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13	Task. Find a non-affine polynomial $u$ (equivalently, one of degree at least 3) in $n \ge 8$ variables, with real algebraic coefficients, satisfying this identity; or prove that every polynomial solution is affine.
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15	## Background
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17	Bernstein proved that entire minimal graphs over $\mathbb{R}^2$ are planes, and through the work of De Giorgi, Almgren and Simons the same was established for $n \le 7$ [2]. Bombieri, De Giorgi and Giusti then constructed non-affine entire minimal graphs over $\mathbb{R}^n$ for every $n \ge 8$ [1]. Their solutions are not polynomials. Simon raised the question of polynomial solutions in his survey of the minimal surface equation [3], which makes it an algebraic version of the Bernstein problem. A positive answer would give the first minimal graph of a non-affine polynomial and might help to build minimal hypersurfaces in high dimensions.
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19	## What is known
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21	- For $n \le 7$ every entire solution is affine, so the question is open only for $n \ge 8$ [2].
22	- Non-affine entire solutions exist for $n \ge 8$, but none known is a polynomial [1].
23	- No non-affine polynomial solution is known in any dimension (Epoch).
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25	## What would count as progress
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27	- An explicit polynomial: its terms with exponent vectors and coefficients in a number field, specified by an irreducible minimal polynomial and an isolating interval for the chosen real root. Substituting into the identity and expanding is exact symbolic computation.
28	- Epoch scores submissions with its own verifier, which receives $n \ge 8$, the coefficient field and the list of terms, and checks that the expression above is identically zero.
29	- A proof that polynomial solutions must be affine would answer Simon's question in the negative.
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31	## References
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33	1. E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. [doi:10.1007/BF01404309](https://doi.org/10.1007/BF01404309)
34	2. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62–105. [doi:10.2307/1970556](https://doi.org/10.2307/1970556)
35	3. L. Simon, The minimal surface equation, in Encyclopaedia of Mathematical Sciences, Springer (1997), 239–266. [doi:10.1007/978-3-662-03484-2_5](https://doi.org/10.1007/978-3-662-03484-2_5)
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37	## Source
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39	Imported from [FrontierMath: Open Problems, polynomial-minimal-surface](https://epoch.ai/frontiermath/open-problems/polynomial-minimal-surface), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.