Complexity Classes and Completeness in Algebraic Geometry

TL;DR

This paper explores the computational complexity of algebraic geometry objects, establishing NP-completeness results for certain geometric problems, and introduces algebraic analogues of classical complexity classes.

Contribution

It defines complexity classes P and NP for algebraic geometry and proves NP-completeness of the universal circuit resultant in this setting.

Findings

Universal circuit resultant is NP-complete in algebraic geometry

First geometric family shown to be NP-complete

Introduces algebraic analogues of P and NP classes

Abstract

We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting.