 
Summary: Improved lowdegree testing and its applications
Sanjeev Arora \Lambda
Princeton University
Madhu Sudan y
IBM Watson Research Center
Abstract
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 [RS93]. The strongest previously known connection
for this test states that a function passes the test with probability ffi for some ffi ? 7=8
iff the function has agreement ß ffi with a polynomial of degree d. We present a new,
and surprisingly strong, analysis which shows that the preceding statement is true for
ffi ø 0:5. The analysis uses a version of Hilbert irreducibility, a tool of algebraic geometry.
As a consequence we obtain an alternate construction for the following proof sys
tem: A constant prover 1round proof system for NP languages in which the verifier
uses O(logn) random bits, receives answers of size O(logn) bits, and has an error prob
ability of at most 2 \Gamma log 1\Gammaffl n . Such a proof system, which implies the NPhardness of
approximating Set Cover to
within\Omega\Gammathi n) factors, has already been obtained by Raz
