Polynomial functions as splines

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).

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