## Abstract

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 double-struck F _{qn} to the subfield double-struck F_{q} and include all properties that form an double-struck F_{q}-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 double-struck F_{q} has a single-orbit characterization (even when the domain is viewed as double-struck F_{qn} 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 double-struck F _{q} 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.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings |

Pages | 400-411 |

Number of pages | 12 |

DOIs | |

State | Published - 2011 |

Externally published | Yes |

Event | 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 - Princeton, NJ, United States Duration: 17 Aug 2011 → 19 Aug 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6845 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 |
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Country/Territory | United States |

City | Princeton, NJ |

Period | 17/08/11 → 19/08/11 |

## Keywords

- Direct sums
- Error-correcting codes
- Property testing
- Symmetries