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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn
% 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu
% HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr
% fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9
% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaaIYaWa
% aWbaaSqabeaacqGHsislciGGSbGaai4BaiaacEgadaahaaadbeqaai
% aaigdacqGHsislcqGHiiIZaaWccaWGUbaaaaaa!4CB1!
 $$
2^{{ - \log ^{{1 - \in }} n}} 
$$
 . 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 % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn
% 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu
% HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr
% fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9
% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacqaH0oaz
% cqGH9aqpcaaIXaGaai4lamaaemaabaGaamOraaGaay5bSlaawIa7am
% aaCaaaleqabaGaeyicI4maaOGaeSOAI0JaaGimaiaac6cacaaI1aaa
% aa!50F9!
 $$
\delta = 1/{\left| F \right|}^{ \in } \ll 0.5
$$
 , then the
tester/corrector determines δ and generates % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn
% 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu
% HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr
% fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9
% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGpbWa
% aeWaaeaadaWcaaqaaiaaigdaaeaacqaH0oazaaaacaGLOaGaayzkaa
% aaaa!4880!
 $$
O{\left( {\frac{1}
{\delta }} \right)}
$$
 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 
% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn
% 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu
% HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr
% fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9
% pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv
% e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGpbWa
% aeWaaeaadaWcaaqaaiaaigdaaeaacqaH0oazaaaacaGLOaGaayzkaa
% aaaa!4880!
 $$
O{\left( {\frac{1}
{\delta }} \right)}
$$
 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.

Improved low-degree testing and its applications

Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. It is shown that within this evaluation-box representation, the polynomial greatest common divisor and factorization problems as well as the problem of extracting the numerator and denominator of a rational function can be solved in random polynomial time in the usual parameters. Since the resulting evaluation programs for the goal polynomials can be converted efficiently to sparse format, solutions to sparse problems such as the sparse ration interpolation problem follow as a consequence. >

Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators

Fast parallel absolute irreducibility testing

We develop an algorithm for the factorization of a multivariate polynomial represented by a straight-line program into its irreducible factors represented as sparse polynomials. Our algorithm is in random polynomial-time for the usual coefficient fields and outputs with controllably high probability the correct factorization. It only requires an a priori bound for the total degree of the input and over rational numbers a bound on the size of the polynomial coefficients.

Computing with polynomials given by straight-line programs II sparse factorization

Consider a polynomial f with an arbitrary but fixed number of variables and with integral coefficients. We present an algorithm which reduces the problem of finding the irreducible factors off in polynomial-time in the total degree off and the coefficient lengths off to factoring a univariate integi'al polynomial. Together with A. Lenstra's,.H. Lenstra's and L. Lovfisz' polynomial-time factorization algorithm for univariate integral polynomials (Math. Ann., 261 (1982), pp. 515-534) this algorithm implies the following theorem. Factoring an integral polynomial with a fixed number of variables into irreducibles, except for the constant factors, can be accomplished in deterministic polynomial-time in the total degree and the size of its coefficients. Our algorithm can be generalized to factoring multivariate polynomials with coefficients in algebraic number fields and finite fields in polynomial-time. We also present a different algorithm, based on an effective version ofa Hilbert IrreducibilityThebrem, which polynomial-time reduces testing multivariate polynomials for irreducibility to testing bivariate integral polynomials for irreduciblity.

Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization*

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Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization*