Workshop: Ranks of matrices with polynomial entries, Igor Shparlinski

Date: 2024-01-22

Time: 09:30 - 10:30


Igor Shparlinski, University of New South Wales


We outline some recent results from a joint work with Ali Mohammadi and Alina Ostafe on statistics of
m x n matrices of the form (f_{i,j}(x_{i,j})) with polynomials f_{ij} \in Z[X] and  |x _{ij}| \le  H and of prescribed rank r. We consider this question over Z and over F_p. Besides being of independent interest, this has also appeared in the recent work of Valentin Blomer and Junxian Li (2022) on correlations of values of random diagonal forms.

We explain the approach of Valentin Blomer and Junxian Li and our improvement of this approach. In turn,  this leads us to a question of counting integer points on some hypersurfaces.

In the Z-case we treat this via the mean value theorems of Hua Loo Keng (1938) and Trevor Wooley (2019). In the F_p-case we use a result of Etienne Fouvry (2000) and a recent bound on the additive energy of polynomials due to Bryce Kerr, Ali Mohammadi and the speaker (2023)

Some open questions of both analytic and algebraic geometry nature will be outlined as well.