The intrinsic complexity of parametric elimination methods

TL;DR

This paper proves that all known symbolic parametric polynomial solving methods with algebraic robustness inherently require exponential time, highlighting fundamental complexity limitations in geometric elimination techniques.

Contribution

It establishes that any algebraically robust parametric elimination method must have exponential sequential time complexity, revealing a fundamental computational barrier.

Findings

All algebraically robust parametric elimination methods are exponentially complex.

The complexity lower bound applies to all known symbolic geometric elimination techniques.

This result clarifies the intrinsic difficulty of parametric polynomial solving with robustness properties.

Abstract

This paper is devoted to the complexity analysis of a particular property, called "algebraic robustness" owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which owns this property must neccessarily have an exponential sequential time complexity.