On Sums of Locally Testable Affine Invariant Properties
نویسندگان
چکیده
Affine-invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field Fqn to the subfield Fq and include all properties that form an Fq-vector space and are invariant under affine transformations of the domain. Almost all the known locally testable affine-invariant properties have so-called “single-orbit characterizations” — namely they are specified by a single local constraint on the property, and the “orbit” of this constraint, i.e., translations of this constraint induced by affine-invariance. Single-orbit characterizations by a local constraint are also known to imply local testability. Despite this prominent role in local testing for affine-invariant properties, single-orbit characterizations are not well-understood. In this work we show that properties with single-orbit characterizations are closed under “summation”. Such a closure does not follow easily from definitions, and our proof uses some of the rich developing theory of affine-invariant properties. To complement this result, we also show that the property of being an n-variate low-degree polynomial over Fq has a singleorbit characterization (even when the domain is viewed as Fqn and so has very few affine transformations). This allows us to exploit known results on the single-orbit characterizability of “sparse” affine-invariant properties to show the following: The sum of any sparse affineinvariant property (properties satisfied by q-functions) with the set of degree d multivariate polynomials over Fq has a single-orbit characterization (and is hence locally testable) when q is prime. Our result leads to the broadest known family of locally testable affine-invariant properties and gives rise to some intriguing questions/conjectures attempting to classify all locally testable affine-invariant properties.
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