Polynomial functions as splines

David Kazhdan; Tamar Ziegler

doi:10.1007/s00029-019-0476-9

Polynomial functions as splines

David Kazhdan^*, Tamar Ziegler

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

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).

Original language	American English
Article number	31
Journal	Selecta Mathematica, New Series
Volume	25
Issue number	2
DOIs	https://doi.org/10.1007/s00029-019-0476-9
State	Published - 1 Jun 2019

Bibliographical note

Funding Information:
Acknowledgements 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.

Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

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N2 - 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).

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Polynomial functions as splines

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