Cubic Curves in F_p^2

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a prime $p$, let $\mathbb F_p$ be the field with $p$ elements. An affine line in $\mathbb F_p^2$ is a set $\{v+t w:t\in\mathbb F_p\}$ with $w\ne0$. A cubic curve here is the zero set of a nonzero polynomial $F\in\mathbb F_p[X,Y]$ of total degree at most three; reducible polynomials are permitted. A curve of smaller positive degree can also be included in a degree-three zero set by multiplying its defining polynomial by linear factors.

Statement recorded in the supplied source.

For every real $\epsilon>0$, is there $p_0(\epsilon)$ such that for every prime $p\geq p_0(\epsilon)$ and every set $A\subset\mathbb F_p^2$ satisfying $|A\cap\ell|\leq2$ for every affine line $\ell$, some nonzero polynomial $F\in\mathbb F_p[X,Y]$ of total degree at most three satisfies \[ |\{a\in A:F(a)\ne0\}|\leq\epsilon p? \] Thus the exceptional $o(p)$ bound must be uniform over the sets $A$, as $p$ tends to infinity through primes. [S2]

Short English statement

Does every planar set over a large prime field with no three points on a line lie almost entirely on a curve of degree at most three?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 129 (GREEN-041). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, maintained author collection, Problem 68 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.