# Colloquium Prof. Sudhir Ghorpade, IIT Bombay: "A Finite Field Nullstellensatz and the Footprint Bound"

Jun 06, 2019 from 02:15 PM to 03:15 PM

CAP 4 - Raum 910

Abstract:

The *Nullstellensatz *or the Zero-Point Theorem is a result of fundamental importance proved by David Hilbert in 1893. It holds when the base field is algebraically closed. In this talk we will first discuss some Nullstellensatz-like results when the base field is finite, and outline the proofs. We will then consider related questions concerning an explicit determination of vanishing ideals of affine as well as projective spaces over finite fields, and the determination of reduced Gröbner bases of these ideals. Next, we shall discuss the so called Footprint bounds from Gröbner basis theory in the context of the number of points of affine or projective algebraic varieties over finite fields. As an application, we will show how the classical bound of Øystein Ore for affine hypersurfaces and of J.-P. Serre for projective hypersurfaces can be derived as a consequence.

Parts of this talk are based on a joint work with Peter Beelen and Mrinmoy Datta.

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