Software Engineering and Complexity in Effective Algebraic Geometry

TL;DR

This paper introduces a new computation model for algebraic geometry problems that captures existing algorithms and demonstrates some problems require exponential time even with efficient algorithms.

Contribution

It presents a robust parameterized arithmetic circuit model for algebraic geometry and justifies it through software engineering principles.

Findings

The model captures all known symbolic algorithms in effective Algebraic Geometry.

Some elimination problems inherently require exponential time.

The model is adapted to Scientific Computing applications.

Abstract

We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures all known branching parsimonious symbolic algorithms in effective Algebraic Geometry. We justify this model by arguments from Software Engineering. Finally we exhibit a class of simple elimination problems of effective Algebraic Geometry which require exponential time to be solved by branching parsimonious algorithms of our computation model.