Polynomial functions as splines
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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 \(f:A \rightarrow k\) 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).
Mathematics Subject Classification05E15 14D05 11B30
The second author is supported by ERC grant ErgComNum 682150. Part of the material in this paper is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. We thank the anonymous referee for offering simplified proofs for some Lemmas in the paper.
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