# 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.

## 6 thoughts on “Related papers”

1. Frank de Zeeuw |

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

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? 🙂

• Frank de Zeeuw |

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.
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.