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.