Improved low-degree testing and its applications

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. The strongest previously known connection for this test states that a function passes the test with probability delta for some delta >7/8 iff the function has agreement approximately delta with a polynomial of degree d. We present a new, and surprisingly strong, analysis which shows that the preceding statement is true for delta much smaller than 1/2. The analysis uses a version of {\em Hilbert irreducibility}, a tool of algebraic geometry. As a consequence we obtain an alternate construction for the following proof system: A constant prover 1-round proof system for NP languages in which the verifier uses O(log n) random bits, receives answers of size O(log n) bits, and has an error probability of at most $2^{-\log^{1 - \epsilon} n}$. Such a proof system, which implies the NP-hardness of approximating Set Cover to within Omega(log n) factors, has already been obtained by Raz and Safra. Our result was completed after we heard of their claim. A second consequence of our analysis is a self tester/corrector for any buggy program that (supposedly) computes a polynomial over a finite field. If the program is correct only on delta fraction of inputs where delta is much smaller than 1/2, then the tester/corrector determines delta and generates O(1/delta) randomized programs such that one of them is correct on every input, with high probability.