On the intrinsic complexity of elimination problems in effective Algebraic Geometry

TL;DR

This paper demonstrates that elimination problems in effective algebraic geometry inherently require exponential complexity, even when using a circuit-based computation model that encompasses all known algorithms.

Contribution

It introduces a circuit-based model capturing all symbolic elimination algorithms and proves the intrinsic exponential complexity of elimination within this framework.

Findings

Elimination in effective algebraic geometry is intrinsically exponential.

The circuit-based model captures all known symbolic elimination algorithms.

Complexity lower bounds are established for elimination problems.

Abstract

The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation model which captures all known symbolic elimination algorithms in effective algebraic geometry and show the intrinsically exponential complexity character of elimination in this complexity model.