MGSC Website: http://math.uci.edu/~mgsc/
We investigate a series of related problems in the area of incomplete Weil sums where the sum is run over a set of points that produces the image of the polynomial. We establish a bound for such sums, and establish some numerical evidence for a conjecture that this sum can be bounded in a way similar to Weil's bounding theorem.
To aide in the average case, we investigate the problem of the cardinality of the value set of a positive degree polynomial (degree $d > 0$) over a finite field with $p^m$ elements. We show a connection between this cardinality and the number of points on a family of varieties in affine space. We couple this with Lauder and Wan's $p$-adic point counting algorithm, resulting in a non-trivial algorithm for calculating this cardinality in the instance that $p$ is sufficiently small.
Josh likes finite fields and long walks on the beach. While not doing math with finite fields on the beach, he reclines on a throne made of math books and gestures with a licorice pipe.
Daqing Wan is currently Joshua's advisor.
none
Pizza will be served after the talk.