Self-testing/correcting with applications to numerical problems
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Checking computations in polylogarithmic time
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Efficient checking of polynomials and proofs and the hardness of approximation problems
Efficient checking of polynomials and proofs and the hardness of approximation problems
Nearly-linear size holographic proofs
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Journal of the ACM (JACM)
On the efficiency of local decoding procedures for error-correcting codes
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Testing Polynomials over General Fields
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Testing Low-Degree Polynomials over Prime Fields
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Simple PCPs with poly-log rate and query complexity
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Some 3CNF Properties Are Hard to Test
SIAM Journal on Computing
A combinatorial characterization of the testable graph properties: it's all about regularity
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Graph limits and parameter testing
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Locally testable codes and PCPs of almost-linear length
Journal of the ACM (JACM)
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Low-degree tests at large distances
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Towards 3-query locally decodable codes of subexponential length
Journal of the ACM (JACM)
Algebraic property testing: the role of invariance
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
2-Transitivity Is Insufficient for Local Testability
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
3-query locally decodable codes of subexponential length
Proceedings of the forty-first annual ACM symposium on Theory of computing
Locally Testable Codes Require Redundant Testers
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Optimal Testing of Reed-Muller Codes
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Symmetric LDPC Codes are not Necessarily Locally Testable
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
IEEE Transactions on Information Theory
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Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, "algebraic property testing" is not well-understood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "affine-invariant properties", i.e., properties that are closed under summation, and under affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its affine-invariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of affine-invariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of Reed-Muller codes (the most popular form of locally testable codes, which also happen to be affine-invariant) could be found by systematically exploring the space of affine-invariant properties. In this work we rule out this possibility and show that general (linear) affine-invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity. A central impediment to proving such results was the limited understanding of the structural restrictions on affine-invariant properties imposed by the existence of local tests. We manage to overcome this limitation and present a clean restriction satisfied by affine-invariant properties that exhibit local tests. We do so by relating the problem to that of studying the set of solutions of a certain nice class of (uniform, homogenous, diagonal) systems of multivariate polynomial equations. Our main technical result completely characterizes (combinatorially) the set of zeroes, or algebraic set, of such systems.