The Szemerédi-Trotter theorem in the complex plane

Tóth, Csaba D.

doi:10.1007/s00493-014-2686-2

Cited by 56 publications

(49 citation statements)

References 21 publications

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“…This is Toth's complex generalization of the Szemerédi-Trotter theorem [Toth03]. If L is a set of L complex lines in C 2 , Toth proved that |S r (L)| L 2 r −3 + Lr −1 -the same estimate as for real lines in R 2 .…”

Section: Some Methods and Problems In Incidence Geometrymentioning

confidence: 62%

Unexpected Applications of Polynomials in Combinatorics

Guth

2013

The Mathematics of Paul Erdős I

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In the last six years, several combinatorics problems have been solved in an unexpected way using high degree polynomials. The most well-known of these problems is the distinct distance problem in the plane. In [Erdős46], Erdős asked what is the smallest number of distinct distances determined by n points in the plane. He noted that a square grid determines ∼ n(log n) −1/2 distinct distances, and he conjectured that this is sharp up to constant factors. Recently, an estimate was proven which is sharp up to logarithmic factors. The main new thing in the proof is the use of high-degree polynomials. This new technique first appeared in Dvir's paper [Dvir09], which solved the finite field Nikodym and Kakeya problems. Experts had considered these problems very difficult, but the proof was essentially one page long. The method has had several other applications. The joints problem was resolved in . The argument was simplified and generalized in [KSS10] and [Quilodrán10], leading to another one page proof. A higher-dimensional generalization of the Szemerédi-Trotter theorem was proven in . And several fundamental theorems in incidence geometry were reproved in the paper [KMS12].The new trick in these proofs can be summarized as follows. We want to understand some finite set S in a vector space. We consider a minimal degree (non-zero) polynomial that vanishes on the set S. Then we use this polynomial to study the problem. This strategy is somewhat surprising because the statements of the problems often involve only points and lines. The joints problem and the finite field Nikodym problem can be solved in a page each using high degree polynomials but seem very difficult to solve without polynomials. Why polynomials play such a crucial role in these problems is somewhat mysterious.The point of this essay is to explain how these new methods work and to reflect on them philosophically. The main theme is the connection between combinatorics and algebra (polynomials).Here is an outline of the essay. We begin by giving two detailed examples of the polynomial method: the finite field Nikodym problem and the joints problem. This is the subject of Section 1: Examples of the polynomial method.Once we've seen a couple examples of this method, we're going to work on understanding "where it comes from". In Section 2, we discuss where the method comes from historically.

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Section: Some Methods and Problems In Incidence Geometrymentioning

confidence: 62%

Unexpected Applications of Polynomials in Combinatorics

Guth

2013

The Mathematics of Paul Erdős I

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“…The known sum-product results over the real or complex field are somewhat stronger than in fields of positive characteristic largely due to order properties of reals, which so far have been indispensable for proofs of the celebrated Szemerédi-Trotter theorem in the plane. See [30], [32] for the original proof for reals and subsequent extension to the complex field.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the energy variant of the sum-product conjecture

2019

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We prove new exponents for the energy version of the Erdős-Szemerédi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the "few sums, many products" inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of the art sum-product exponent over the reals due to Konyagin and the second author, up to 4 3 + 1 1509 . An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.

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“…This section has two parts. First, we develop an application of Theorem 1 to the problem of counting distinct values of a non-degenerate bilinear form on pairs of points lying in a plane point set, extending to positive characteristic the estimates one easily obtains over R (as well as C, for it applies there as well [48]) via the Szemerédi-Trotter theorem. Then we consider the positive characteristic version of the Erdős distance problem in dimensions three and two.…”

Section: Applications To Erdős-type Geometric Questionsmentioning

confidence: 99%

11. Point-plane incidences and some applications in positive characteristic

Rudnev¹

2019

Combinatorics and Finite Fields

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The point-plane incidence theorem states that the number of incidences between n points and m ≥ n planes in the projective three-space over a field F , iswhere k is the maximum number of collinear points, with the extra condition n < p 2 if F has characteristic p > 0. This theorem also underlies a state-of-the-art Szemerédi-Trotter type bound for point-line incidences in F 2 , due to Stevens and de Zeeuw. This review focuses on some recent, as well as new, applications of these bounds that lead to progress in several open geometric questions in F d , for d = 2, 3, 4. These are the problem of the minimum number of distinct nonzero values of a non-degenerate bilinear form on a point set in d = 2, the analogue of the Erdős distinct distance problem in d = 2, 3 and additive energy estimates for sets, supported on a paraboloid and sphere in d = 3, 4. It avoids discussing sum-product type problems (corresponding to the special case of incidences with Cartesian products), which have lately received more attention.2000 Mathematics Subject Classification. 68R05,11B75.1 the folklore that the proof should work, with some constraints, over a general field. The first "official" account of this was given by Ellenberg and Hablicsek [16] in late 2013, followed by Kollár [28] and the author [39] in 2014. The latter two had been aware of a 2003 paper by Voloch [50], which discussed the constraints under which the key element of the First Guth-Katz theorem proof, the Monge-Salmon theorem [42], applied in positive characteristic.Theorem 2 (First Guth-Katz theorem). Let L be a set of lines in C 3 . Suppose, no more then two lines are concurrent. Then the number of pair-wise intersections of lines in L is O |L| 3 2 + |L|k ,

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The Szemerédi-Trotter theorem in the complex plane

Cited by 56 publications

References 21 publications

Unexpected Applications of Polynomials in Combinatorics

Unexpected Applications of Polynomials in Combinatorics

On the energy variant of the sum-product conjecture

11. Point-plane incidences and some applications in positive characteristic

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