# Detecting Rational Points on Hypersurfaces over Finite Fields

```@article{Kopparty2008DetectingRP,
title={Detecting Rational Points on Hypersurfaces over Finite Fields},
author={Swastik Kopparty and Sergey Yekhanin},
journal={2008 23rd Annual IEEE Conference on Computational Complexity},
year={2008},
pages={311-320}
}```
We study the complexity of deciding whether a given homogeneous multivariate polynomial has a non- trivial root over a finite field. Given a homogeneous algebraic circuit C that computes an n- variate polynomial p(x) of degree d over a finite field Fq, we wish to determine if there exists a nonzero xisinFq n with C(x)=0. For constant n there are known algorithms for doing this efficiently. However for linear n, the problem becomes NP hard. In this paper, using interesting algebraic techniques… CONTINUE READING