The purpose of this page is to list the papers that are related to the recent polynomial method / algebraic method.

Legend

- – related to partitioning polynomials.
- – related to incidence problems.
- – related to additive combinatorics.
- – related to distinct distances problems.
- – related to the Elekes-Sharir-Guth-Katz framework, or to its partial variant.

## The papers:

- “On the size of Kakeya sets in finite fields” by Dvir.
- “Algebraic methods in discrete analogs of the Kakeya problem” by Guth and Katz.
- “On lines and Joints” by Kaplan, Sharir, and Shustin.
- “The joints problem in “ by Quilodrán.
- “On Lines, Joints, and Incidences in Three Dimensions” by Elekes, Kaplan, and Sharir ().
- “Incidences in Three Dimensions and Distinct Distances in the Plane” by Elekes and Sharir (,,).
- “On the Erdos distinct distance problem in the plane” by Guth and Katz (,,,).
- “Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique” by Kaplan, Matoušek, and Sharir (,).
- “An improved bound on the number of point-surface incidences in three dimensions” by Zahl (,).
- “On an application of Guth-Katz theorem” by Iosevich, Roche-Newton, and Rudnev (,).
- “An incidence theorem in higher dimensions” by Solymosi and Tao (,).
- “Unit Distances in Three Dimensions” by Kaplan, Matoušek, Safernova, and Sharir (,).
- “Three-point configurations determined by subsets of via the Elekes-Sharir paradigm” by Bennett, Iosevich, and Pakianathan (,).
- “A Szemeredi-Trotter type theorem in “ by Zahl (,).
- “On the Minkowski distances and products of sum sets” by Roche-Newton and Rudnev (,).
- ” Counting joints with multiplicities” by Iliopoulou ().
- “On the number of classes of triangles determined by points in “ by Rudnev ().
- “Improved bounds for incidences between points and circles” by Sharir, Sheffer, and Zahl (,).
- “On Range Searching with Semialgebraic Sets II “ by Agarwal, Matoušek, and Sharir ().
- “A note on distinct distance subsets” by Charalambides ().
- “Distinct distances on two lines” by Sharir, Sheffer, and Solymosi (,,).
- “On lattices, distinct distances, and the Elekes-Sharir framework” by Cilleruelo, Sharir, and Sheffer (,).
- “Distinct distances from three points” by Sharir and Solymosi (,,).
- “Exponent gaps on curves via rigidity” by Charalambides ().
- “Distinct Distances on Algebraic Curves in the Plane” by Pach and de Zeeuw (,,).
- “Few distinct distances implies no heavy lines or circles” by Sheffer, Zahl, and de Zeeuw (,,,).
- “Bounds of incidences between points and algebraic curves” by Wang, Yang, and Zhang (,).

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Nice list!

One suggestion: “Counting joints with multiplicities” by Iliopoulou.

Thanks Frank! I was not aware of this paper. I’m probably missing something, since it seems to me that the result is an immediate corollary of the Elekes-Kaplan-Sharir result (see link above). If there are lines and joints, then we have . The new sum is the number of line-joint incidences, and by replacing it with the Elekes-Kaplan-Sharir bound, we immediately obtain the same bound as in the paper. Can you see what I am missing? 🙂

I think it´s that is the number of triples of lines through that form a joint, so is not the number of line-joint incidences. Although I’m not sure which bound from EKS you mean, so maybe it’s me that’s missing something.

See also the journal version, where there is an added remark about EKS after Theorem 1.1.

Actually, what I found most interesting was Section 4, which is a bit understated in the Introduction. There she proves the same thing for algebraic curves, so where a joint is a triple intersection point with linearly independent tangent vectors.

You’re right. I had the wrong definition of in mind. Then this result does not seem to be a corollary of Elekes-Kaplan-Sharir, though I can think of a similar corollary. Say that is the number of joints through which there are at least lines. Then we have . Section 4 does look very interesting! I’ll definitely try to go over it soon.

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