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Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma forsemi-algebraic $k$-uniform hypergraphs of bounded complexity, showing that foreach $\epsilon>0$ the vertex set can be equitably partitioned into a boundednumber of parts (in terms of $\epsilon$ and the complexity) so that all but an$\epsilon$-fraction of the $k$-tuples of parts are homogeneous. We prove thatthe number of parts can be taken to be polynomial in $1/\epsilon$. Our improvedregularity lemma can be applied to geometric problems and to the followinggeneral question on property testing: is it possible to decide, with querycomplexity polynomial in the reciprocal of the approximation parameter, whethera hypergraph has a given hereditary property? We give an affirmative answer fortesting typical hereditary properties for semi-algebraic hypergraphs of boundedcomplexity.

A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing