NP = PCP(log n, 1) and related results crucially depend upon the close connection between the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [30]. The strongest previously known connection for this test states that a function passes the test with probability δ for some δ > 7/8 iff the function has agreement ≈ δ with a polynomial of degree d. We present a new, and surprisingly strong, analysis which shows that the preceding statement is true for arbitrarily small ≈, provided the field size is polynomially larger than d/δ. The analysis uses a version of Hilbert irreducibility, a tool of algebraic geometry.
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