Bounds on image sets of planar functions

Time: 9:30 -- 11: 00, July 09, 2025

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: planar function $f$ is a polynomial over a finite field with the property that for every $ainmathbb F_q^*,$ the function $f(x+a)-f(x)$ has a full image. One longstanding question in the intersection of finite geometry, cryptography, and finite fields is the upper bound of the size of the image set of a planar function. We give the best known upper bounds via an almost completely combinatorial argument. However, we believe that our bound can be improved, as if it is tight, this would imply the existence of a projective plane of order 18, violating the prime-power conjecture for projective planes. We hope to eventually see tighter bounds that exploit the underlying algebraic structures, which we have not yet been able to use.