# On the Complexity of Computing the Topology of Real Algebraic Space Curves

## TL;DR

This paper introduces a deterministic algorithm for analyzing the topology of algebraic space curves, achieving the most efficient complexity bound to date, which improves upon previous results significantly.

## Contribution

The paper presents a new deterministic algorithm for computing the topology of algebraic space curves with the best known complexity bound, improving previous methods.

## Key findings

- Complexity bound of $ ilde{O}(N^{20})$ established.
- Algorithm finds a strong generic position for algebraic space curves.
- The new bound improves previous results by at least a factor of $N^2$.

## Abstract

In this paper, we present a deterministic algorithm to find a strong generic position for an algebraic space curve. We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of the algorithm. It is $\tilde{\mathcal {O}} (N^{20})$, where $N=\max\{d,\tau\}$, $d, \tau$ are the degree bound and the bit size bound of the coefficients of the defining polynomials of the algebraic space curve. To our knowledge, this is the best bound among the existing work. It gains the existing results at least $N^2$.