Incidences in a Recent Work of Walsh

Recently, Miguel Walsh posted a very interesting paper on arXiv. The main purpose of the paper is to study various properties of polynomials and varieties. These properties are related to incidence problems – some originally arose from studying incidences. Walsh also presents new incidence bounds as applications of his results. In this post I’ll briefly discuss the incidence aspects of Walsh’s paper. Since this is a rather specialized post for incidence researchers, it assumes familiarity with the problems and notation.

walsh
Miguel Walsh

It seems to me that Walsh’s paper makes three main contributions to the study of incidence problems. The first two contributions are explicitly stated in the paper:

  • The paper derives a new polynomial partitioning theorem for points on a variety, with a good dependency on the degree of the variety. This was the missing step for removing an extra \varepsilon  in the exponent from most of the existing incidence bounds in {\mathbb R}^d  . For example, it should remove the extra \varepsilon  in the results of several of my papers, such as this one and that one.
  • As an application of his results, Walsh introduces an incidence problem that I did not see before. In this problem we study incidences between points and varieties in {\mathbb R}^d  , but the degrees of the varieties are not constant. That is, the degrees of the varieties are allowed to depend on the number of points and on the number of varieties. It would be interesting to see how this variant develops. For example, what applications in other subfields it might have.

The two above incidence results are definitely interesting, but I suspect that the paper would have a more significant effect on the incidence community in a different way. The paper makes significant progress in our understanding of polynomials and varieties, which directly affect the algebraic methods used to derive incidence bounds. This improved understanding could potentially be what we need to address some of the main problems that we have been trying to solve for a while. Me and other incidences researchers will definitely spend time reading the paper carefully, and think how it might fit with these problems. Walsh also mentions a follow-up paper, and I wonder if he has something similar in mind. I am planning to soon write a post about the current main open problems in incidence theory, and will then mention more clearly what I am referring to.

It might also be worth discussing the lower bounds part of the paper. The paper contains matching lower bounds for some of its incidence results. However, these lower bounds seem to apply specifically to the new variant of incidences with varieties of non-constant degrees. The “standard” problem of incidences with bounded-degree hypersurfaces in {\mathbb R}^d  remains open. In particular, we have matching lower bounds only in some cases. For an up-to-date discussion of the known lower bounds, see for example the relevant part in the introduction of my recent work with Thao Do.

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