Zeta function of projective hypersurfaces with ADE singularities

TL;DR

This paper presents a Sage algorithm for computing the zeta function of 3D projective hypersurfaces with ADE singularities, extending previous work and improving efficiency through a novel polynomial equivalence approach.

Contribution

Introduces a new Sage-based algorithm for hypersurfaces with ADE singularities, establishing polynomial equivalence to enhance computational efficiency.

Findings

Successfully computed zeta functions for hypersurfaces with ADE singularities.

Established a polynomial equivalence that simplifies the differential operator conditions.

Enhanced algorithm performance over previous methods.

Abstract

Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacobian ideal with a polynomial satisfying a set of differential operators.