Lately I did not have much time to survey the recent developments in the polynomial method / algebraic combinatorial geometry / whatever it is called these days. Hopefully this will change soon, but for now here is a list of papers from the past year that were not discussed in this blog before. I think that these illustrate how active the field currently is (and I did not include several recent results that have been announced but are not available online yet). Feel free to point out other papers which I might have missed.
1. Polynomial partitioning for a set of varieties / Guth. This paper introduces a new type of polynomial partitioning. Instead of defining a partitioning with respect to a point set such that no cell contains a bounded number of points, this partitioning is defined with respect to a set of varieties and satisfies that every cells intersects a bounded number of varieties. It would be interesting to find new applications for such partitionings.
2. On the use of Klein quadric for geometric incidence problems in two dimensions / Rudnev and Selig. Studies the Elekes-Sharir(-Guth-Katz) framework from a somewhat new perspective. The paper generalizes several variants of the framework, bypassing the standard use of symmetry in such cases.
3. On the number of rich lines in truly high dimensional sets / Dvir and Gopi. The paper derives an upper bound for the number of -rich lines in . For this purpose, Dvir and Gopi combine polynomial
partitioning interpolation with a method relying on design matrices. In recent years, this method was used to obtain bounds for various higher dimensional variants of the Sylvester-Gallai problem.
4. On the number of rich lines in high dimensional real vector spaces / Hablicsek and Scherr. Almost immediately after the Gopi-Dvir paper was placed on arXiv, this paper appeared. It improved the bound of Gopi and Dvir in the case of points and lines in .
5. Generalizations of the Szemerédi-Trotter Theorem / Kalia and Yang. The paper studies a new variant of incidence problems. In , we have a set of points, a set of lines, and a set of planes. An incidence is a point-line-plane triple such that the point is contained in the line and the line is contained in the plane. A similar situation holds in where we have a -tuple of flats of distinct dimensions. (A petty comment – I would have chosen a different title, since this one easily fits over a hundred existing papers)
6. Incidences between points and lines in / Sharir and Solomon. This paper improves upon the authors’ previous result concerning point-line incidences in . The in the exponent is removed, and the restriction on the lines is more subtle. Beyond the result, the tools that the authors develop to handle lines in seem quite interesting.
7. Incidences between points and lines in three dimensions / Sharir and Solomon. The paper presents an alternative proof for the Guth-Katz point-line incidence bound in (substituting the Guth-Katz proof for -rich points in the case of , but not for the case of ). I did not read the new proof yet, but the authors state that it relies on more elementary tools than the Guth-Katz proof.
8. Incidences between points and lines on a two-dimensional variety / Sharir and Solomon. Yet another Sharir-Solomon paper dealing with point-line incidences. This time the authors study incidences in where the lines are all contained in a common two-dimensional variety.
9. Counting multijoints / Iliopoulou. This paper studies a variant of the joints problem in . Instead of having only one set of lines, we have three sets of lines, and a joint consists of one line out of each set. The bound that Iliopoulou obtains seems like the natural one, extending the original joints bound. The result is also extended from lines to general algebraic curves.
10. Incidence bounds on multijoints and generic joints / Iliopoulou. This paper is a collection of results concerning joints. The multijoints result from Iliopoulou’s previous paper is extended to any field. Moreover, in the real space it is extended to any dimension . The paper also studies the case where we have a generic set of lines in . This is a set satisfying that when a subset of lines all meet in a point, they must form a joint.
11. On the joints problem with multiplicities / Hablicsek. This paper studies another joints problem in an arbitrary field , this time taking the multiplicity of each joint into account. That is, one point with many lines through it can correspond to many distinct joints.
12. On the number of ordinary circles / Nassajian Mojarrad and de Zeeuw. For a long time I was surprised that no paper extended or relied on the wonderful Green-Tao result concerning ordinary lines. This paper finally does that, extending the result to “ordinary circles”.
13. Szemerédi–Trotter-type theorems in dimension 3 / Kollár. Kollár studies point-line incidences in and in finite fields. He also studies the exact constants that the recent techniques lead to.
14. Bisector energy and few distinct distances / Lund, Sheffer, and de Zeeuw. We define the new concept of bisector energy and rely on it for a couple of applications. One such application is a new approach for trying to characterize planar point sets that span a small number of distinct distances.