Polynomial functions as splines

TL;DR

The paper establishes a local criterion to determine when a function defined on a subset of a vector space over a finite field extends to a polynomial of degree less than m, with applications to high rank hypersurfaces and local testability.

Contribution

It introduces a new local criterion for polynomial extension on subsets of vector spaces over finite fields, applicable to high rank hypersurfaces and demonstrating robustness.

Findings

High rank hypersurfaces satisfy the criterion.

The criterion is locally testable.

Applicable to polynomial functions of degree < m.

Abstract

Let $V$ be a vector space over a finite field $k$. We give a condition on a subset $A \subset V$ that allows for a local criterion for checking when a function $f:A \to k$ is a restriction of a polynomial function of degree $<m$ on $V$. In particular, we show that high rank hypersurfaces of $V$ of degree $\ge m$ satisfy this condition. In addition we show that the criterion is robust (namely locally testable in the theoretical computer science jargon).