Mathematics Graduate Student Colloquium

Weil Image Sums and Counting Image Sets Over Finite Fields

Joshua Hill
Wednesday, October 19, 2011
4:00 pm - 4:50 pm
RH440R

Talk Abstract:

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.

About the Speaker:

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.

Advisor and Collaborators

Daqing Wan is currently Joshua's advisor.

Supplementary Materials:

none

Refreshments:

Pizza will be served after the talk.

Last Modified: October 25, 2011 at 4:23 AM (UTC)
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