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28
The distribution of polynomials over finite fields, with applications to the Gowers norms
, 2007
"... In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P: F n → F is poorlydistributed only if P is determined by the values of a few polynomials of lower ..."
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Cited by 40 (2 self)
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In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P: F n → F is poorlydistributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the socalled Gowers norms. We establish an inverse result for the Gowers U d+1norm of functions of the form f(x) = eF(P(x)), where P: F n → F is a polynomial of degree less than F, showing that this norm can only be large if f correlates with eF(Q(x)) for some polynomial Q: F n → F of degree at most d. The requirement deg(P) < F  cannot be dropped entirely. Indeed, we show the above claim fails in characteristic 2 when d = 3 and deg(P) = 4, showing that the quartic symmetric polynomial S4 in F n 2 has large Gowers U 4norm but does not correlate strongly with any cubic polynomial. This shows that the theory of Gowers norms in low characteristic is not as simple as previously supposed. This counterexample has also been discovered independently by Lovett, Meshulam, and Samorodnitsky [15]. We conclude with sundry other applications of our main result, including a recurrence result and a certain type of nullstellensatz.
Discrepancy and the power of bottom fanin in depththree circuits
 In Proc. of the 48th Symposium on Foundations of Computer Science (FOCS
, 2007
"... We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty ‘Number on the Forehead ’ model. Our method is based on the notion of voting polynomial degree of functions and extends the DegreeDiscrepancy Lemma in the recent wor ..."
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Cited by 26 (2 self)
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We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty ‘Number on the Forehead ’ model. Our method is based on the notion of voting polynomial degree of functions and extends the DegreeDiscrepancy Lemma in the recent work of Sherstov [24]. Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fanin at the base layer i.e. circuits of type MAJ ◦ SYMM ◦ ANY O(1) cannot simulate the circuit class AC 0 in subexponential size. Further, even if the fanin of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC 0 in quasipolynomial size. This is in contrast to the classical result of Yao and BeigelTarui that shows that such circuits, having only MAJORITY gates, can simulate the class ACC 0 in quasipolynomial size when the bottom fanin is increased to polylogarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain [7] for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented by polynomials of small degree over Zm, when m, q ≥ 2 are coprime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of [7, 14]. It is known that such estimates imply that circuits of type MAJ ◦ MODm ◦ ANDɛ log n cannot compute the MODq function in subexponential size. It remains a major open question to determine if such circuits can simulate ACC 0 in polynomial size when the bottom fanin is increased to polylogarithmic size. 1
Inverse Conjecture for the Gowers norm is false
 In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC
, 2007
"... Let p be a fixed prime number, and N be a large integer. The ’Inverse Conjecture for the Gowers norm ’ states that if the ”dth Gowers norm ” of a function f: F N p → F is nonnegligible, that is larger than a constant independent of N, then f can be nontrivially approximated by a degree d − 1 poly ..."
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Cited by 20 (4 self)
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Let p be a fixed prime number, and N be a large integer. The ’Inverse Conjecture for the Gowers norm ’ states that if the ”dth Gowers norm ” of a function f: F N p → F is nonnegligible, that is larger than a constant independent of N, then f can be nontrivially approximated by a degree d − 1 polynomial. The conjecture is known to hold for d = 2, 3 and for any prime p. In this paper we show the conjecture to be false for p = 2 and for d = 4, by presenting an explicit function whose 4th Gowers norm is nonnegligible, but whose correlation any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao [5]. Their analysis uses a modification of a Ramseytype argument of Alon and Beigel [1] to show inapproximability of certain functions by lowdegree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime p, for d = p 2.
Hardness amplification proofs require majority
 In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC
, 2008
"... Hardness amplification is the fundamental task of converting a δhard function f: {0, 1} n → {0, 1} into a (1/2 − ɛ)hard function Amp(f), where f is γhard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, ɛ, δ are small (and δ = 2 −k captures the case where f i ..."
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Cited by 20 (5 self)
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Hardness amplification is the fundamental task of converting a δhard function f: {0, 1} n → {0, 1} into a (1/2 − ɛ)hard function Amp(f), where f is γhard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, ɛ, δ are small (and δ = 2 −k captures the case where f is worstcase hard). Achieving ɛ = 1/n ω(1) is a prerequisite for cryptography and most pseudorandomgenerator constructions. In this paper we study the complexity of blackbox proofs of hardness amplification. A class of circuits D proves a hardness amplification result if for any function h that agrees with Amp(f) on a 1/2 + ɛ fraction of the inputs there exists an oracle circuit D ∈ D such that D h agrees with f on a 1 − δ fraction of the inputs. We focus on the case where every D ∈ D makes nonadaptive queries to h. This setting captures most hardness amplification techniques. We prove two main results: 1. The circuits in D “can be used ” to compute the majority function on 1/ɛ bits. In particular, these circuits have large depth when ɛ ≤ 1/poly log n. 2. The circuits in D must make Ω � log(1/δ)/ɛ 2 � oracle queries. Both our bounds on the depth and on the number of queries are tight up to constant factors.
Optimal testing of ReedMuller codes
, 2009
"... We consider the problem of testing if a given function ..."
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Cited by 20 (9 self)
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We consider the problem of testing if a given function
The correlation between parity and quadratic polynomials mod 3
 J. Comput. System Sci
"... We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fanin two at the inputs must be of s ..."
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Cited by 17 (4 self)
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We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fanin two at the inputs must be of size 2 Ω(n). This is the first result of this type for general mod 3 subcircuits with ANDs of fanin greater than 1. This yields an exponential improvement over a longstanding result of Smolensky, answering a question recently posed by Alon and Beigel. The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai, Green and Thierauf. The exponential sum and correlation bounds presented here are tight. 1
Constructing ramsey graphs from boolean function representations
 In 21st Annual IEEE Conference on Computational Complexity
, 2006
"... Explicit construction of Ramsey graphs or graphs with no large clique or independent set has remained a challenging open problem for a long time. While Erdos's probabilistic argument shows the existence of graphs on vertices with no clique or independent set of size, the best explicit construct ..."
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Cited by 13 (3 self)
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Explicit construction of Ramsey graphs or graphs with no large clique or independent set has remained a challenging open problem for a long time. While Erdos's probabilistic argument shows the existence of graphs on vertices with no clique or independent set of size, the best explicit constructions with vertices achieve only a bound of . Constructing Ramsey graphs is closely related to polynomial representations of Boolean functions; a low degree representation for OR function can be used to explicitly construct Ramsey graphs [Gro00]. We generalize the above relation by proposing a new framework. We propose a new denition of OR representations: a pair of polynomials represent the OR function on if the union of their zero sets contains all points in except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the best known constructions, ones to due to FranklWilson [FW81], Grolmusz [Gro00] and Alon [Alo98] are captured by this framework; they can all be derived from various OR representations of degree ff based on symmetric polynomials. Thus the barrier to better Ramsey constructions through current techniques appears to be the construction of lower degree representations. Using new algebraic techniques, we show that the above Ramsey constructions cannot be improved using symmetric polynomials.
New correlation bounds for GF(2) polynomials using Gowers uniformity
 Electronic Colloquium on Computational Complexity
, 2006
"... We study the correlation between lowdegree GF (2) polynomials p and explicit functions. Our main results are the following: I We prove that the Modm function on n bits has correlation at most exp � −Ω � n/4 d�� with any GF (2) polynomial of degree d, for any fixed odd integer m. This improves on th ..."
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Cited by 8 (4 self)
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We study the correlation between lowdegree GF (2) polynomials p and explicit functions. Our main results are the following: I We prove that the Modm function on n bits has correlation at most exp � −Ω � n/4 d�� with any GF (2) polynomial of degree d, for any fixed odd integer m. This improves on the previous exp � −Ω � n/8 d� � bound by Bourgain (C. R. Acad. Sci. Paris, 2005) and Green et al. (C. R. Acad. Sci. Paris, 2005). II We exhibit a polynomialtime computable function on n bits that has correlation at most exp � −Ω � n/2 d� � with any GF (2) polynomial of degree d. Previous to our work the best correlation bound for an explicit function was exp � −Ω � n / � d · 2 d�� �, which follows from (Chung and Tetali; SIAM J. Discrete Math., 1993). III We derive an ‘XOR Lemma ’ for lowdegree GF (2) polynomials: We show that if a function f has correlation at most 1 − 4 −d with any GF (2) polynomial of degree d (and Prx[f(x) = 1] ≈ 1/2) then the XOR of m independent copies of f has correlation at most exp � −Ω � m/4 d� � with any GF (2) polynomial of degree d. Our results rely on a measure of the ‘complexity ’ of a function due to Gowers (Geom.
Incomplete quadratic exponential sums in several variables
 GR10] [Gre04] [GRS05] [Rad09] [Smo87] Frederic Green and Amitabha
, 2010
"... Abstract. We consider incomplete exponential sums in several variables of the form S(f, n, m) = 1 ..."
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Abstract. We consider incomplete exponential sums in several variables of the form S(f, n, m) = 1
On the correlation between parity and modular polynomials
 Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science
, 2006
"... Abstract. We consider the problem of bounding the correlation between parity and modular polynomials over Zq, for arbitrary odd integer q ≥ 3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bo ..."
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Abstract. We consider the problem of bounding the correlation between parity and modular polynomials over Zq, for arbitrary odd integer q ≥ 3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bounds on the size necessary to compute parity by depth3 circuits of certain form. Our technique is based on a new representation of the correlation using exponential sums. Our results include Goldmann’s result [Go] on the correlation between parity and degree one polynomials as a special case. Our general expression for representing correlation can be used to derive the bounds of Cai, Green, and Thierauf [CGT] for symmetric polynomials, using ideas of the [CGT] proof. The classes of polynomials for which we obtain exponentially small upper bounds include polynomials of large degree and with a large number of terms, that previous techniques did not apply to. 1