Ben-Sasson, EliGrigorescu, ElenaMaatouk, GhidShpilka, AmirSudan, Madhu2013-07-102013-07-102013-07-10201110.1007/978-3-642-22935-0_34https://infoscience.epfl.ch/handle/20.500.14299/93246Affine-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. In this work we show that properties with single-orbit characterizations are closed under “summation”. To complement this result, we also show that the property of being an n-variate low-degree polynomial over Fq has a single-orbit characterization (even when the domain is viewed as Fqn and so has very few affine transformations). As a consequence we find that the sum of any sparse affine-invariant property (properties satisfied by q O(n)-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. We conclude with some intriguing questions/conjectures attempting to classify all locally testable affine-invariant properties.Property testingtext::conference output::conference proceedings::conference paper