Polynomial functions as splines

Let V be a vector space over a finite field k. We give a condition on a subset A⊂ V that allows for a local criterion for checking when a function f: A→ 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 ≥ m satisfy this condition. In addition we show that the criterion is robust (namely locally testable in the theoretical computer science jargon).