Related papers

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

Legend

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

The papers:

6 thoughts on “Related papers

  1. Pingback: New page – Related papers | Some Plane Truths

  2. 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 n lines and m joints, then we have \sum N(x)^{1/2} \le \sqrt{m\sum N(x)}. 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 N(x) is the number of triples of lines through x that form a joint, so \sum N(x) 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.

  3. You’re right. I had the wrong definition of N(x) 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 m_{\ge k} is the number of joints through which there are at least k lines. Then we have \sum N(x) = \sum_k \binom{k}{3} (m_{\ge k} -m_{\ge k+1}) = ... = O(n^3). Section 4 does look very interesting! I’ll definitely try to go over it soon.

  4. Pingback: Recent Results (Feb `14) | Some Plane Truths

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