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Simultaneous prime specializations of polynomials over finite fields
"... Recently the author proposed a uniform analogue of the BatemanHorn conjectures for polynomials with coefficients from a finite field (i.e., for polynomials in Fq[T] rather than Z[T]). Here we use an explicit form of the Chebotarev density theorem over function fields to prove this conjecture in par ..."
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Recently the author proposed a uniform analogue of the BatemanHorn conjectures for polynomials with coefficients from a finite field (i.e., for polynomials in Fq[T] rather than Z[T]). Here we use an explicit form of the Chebotarev density theorem over function fields to prove this conjecture in particular ranges of the parameters. We give some applications including the solution of a problem posed by C. Hall.
TESTING LINEARINVARIANT NONLINEAR PROPERTIES
"... We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for ReedMuller codes, has mostly focused on such tasks for linear properties. The one ex ..."
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We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for ReedMuller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for “triangle freeness”: A function f: F n 2 → F2 satisfies this property if f(x), f(y), f(x + y) do not all equal 1, for any pair x, y ∈ F n 2. Here we extend this test to a more systematic study of testing for linearinvariant nonlinear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by k points v1,..., vk ∈ F k 2 and f: F n 2 → F2 satisfies the property that if for all linear maps L: F k 2 → F n 2 it is the case that f(L(v1)),..., f(L(vk)) do not all equal 1. We show that this property is testable if the underlying matroid specified by v1,..., vk is a graphic matroid. This extends Green’s result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of “1complexity linear systems” of Green and Tao, and graphic matroids, to derive the results.
NORM FORMS FOR ARBITRARY NUMBER FIELDS AS PRODUCTS OF LINEAR POLYNOMIALS
"... Abstract. Given a number field K/Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) = P (t). Combining additive combinatorics with descent we show that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak appr ..."
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Abstract. Given a number field K/Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) = P (t). Combining additive combinatorics with descent we show that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X. Contents
Every locally characterized affineinvariant property is testable
, 2013
"... Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious ..."
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Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious test for any such property P, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies P, and otherwise reject with probability larger than a positive number that depends only on the distance between f and P. More generally, we show that any affineinvariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of being decomposable into a known structure of lowdegree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degreed polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degreestructural properties and are therefore locally characterized. Our results use a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of a small number of lowdegree nonclassical polynomials. We establish a new equidistribution result for high rank nonclassical polynomials that drives the proofs of both the testability results and the local characterization of degreestructural properties.
Correlation testing for affine invariant properties on Fnp in the high error regime
 In Proc. 43rd Annual ACM Symposium on the Theory of Computing
, 2011
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MULTIPLE RECURRENCE AND CONVERGENCE ALONG THE PRIMES
"... Abstract. Let E ⊂ Z be a set of positive upper density. Suppose that P1, P2,..., Pk ∈ Z[X] are polynomials having zero constant terms. We show that the set E ∩(E −P1(p−1))∩...∩(E −Pk(p−1)) is nonempty for some prime number p. Furthermore, we prove convergence in L 2 of polynomial multiple averages ..."
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Abstract. Let E ⊂ Z be a set of positive upper density. Suppose that P1, P2,..., Pk ∈ Z[X] are polynomials having zero constant terms. We show that the set E ∩(E −P1(p−1))∩...∩(E −Pk(p−1)) is nonempty for some prime number p. Furthermore, we prove convergence in L 2 of polynomial multiple averages along the primes. 1.
A relative Szemerédi theorem
"... The celebrated GreenTao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmet ..."
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The celebrated GreenTao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for kterm arithmetic progressions in pseudorandom subsets of ZN of density N −ck. The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemerédi theorem.