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.

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
\( 2^{{ - \log ^{{1 - \in }} n}} \). Such a proof system,
which implies the NP-hardness of approximating Set Cover to
within Ω(log *n*) factors, has
already been obtained by Raz and Safra [29]. Raz and Safra
obtain their result by giving a strong analysis, in the sense
described above, of a new low-degree test that they
present.

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 δ fraction of inputs where \( \delta = 1/{\left| F \right|}^{ \in } \ll 0.5 \), then the tester/corrector determines δ and generates \( O{\left( {\frac{1} {\delta }} \right)} \) values for every input, such that one of them is the correct output. In fact, our results yield something stronger: Given the buggy program, we can construct \( O{\left( {\frac{1} {\delta }} \right)} \) randomized programs such that one of them is correct on every input, with high probability. Such a strong self-corrector is a useful tool in complexity theory—with some applications known.

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## Additional information

* Supported by an NSF CAREER award, an Alfred P. Sloan Fellowship, and a Packard Fellowship.

† Part of this work was done when this author was at the IBM Thomas J. Watson Research Center.

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### Cite this article

Arora*, S., Sudan†, M. Improved Low-Degree Testing and its
Applications.
*Combinatorica* **23, **365–426 (2003). https://doi.org/10.1007/s00493-003-0025-0

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*AMS Subject Classification
(2000):*

- 68Q10
- 68Q17