Abstract
We describe an efficient randomized algorithm to test if a given binary function f: {0,1}n →{0,1} is a low-degree polynomial (that is, a sum of low-degree monomials). For a given integer k ≥ 1 and a given real ε >0, the algorithm queries f at \(O(\frac{1}{\epsilon}+k4^k)\) points. If f is a polynomial of degree at most k, the algorithm always accepts, and if the value of f has to be modified on at least an ε fraction of all inputs in order to transform it to such a polynomial, then the algorithm rejects with probability at least 2/3. Our result is essentially tight: Any algorithm for testing degree-k polynomials over GF(2) must perform \(\Omega(\frac{1}{\epsilon}+2^k)\) queries.
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Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D. (2003). Testing Low-Degree Polynomials over GF(2). In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds) Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. RANDOM APPROX 2003 2003. Lecture Notes in Computer Science, vol 2764. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45198-3_17
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DOI: https://doi.org/10.1007/978-3-540-45198-3_17
Publisher Name: Springer, Berlin, Heidelberg
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