[Paper Review] On the Computation of the Topology of a Non-Reduced Implicit Space Curve

Daouda Niang Diatta, Bernard Mourrain+1 more|Mar 7, 2008

Polynomial and algebraic computation7 references5 citations

TL;DR

This paper presents a certified symbolic-numeric algorithm for computing the exact topology of non-reduced space curves defined as the intersection of two implicit algebraic surfaces. Using a single projection onto the (x,y)-plane and rational parametrization via subresultants, it constructs a piecewise linear structure isotopic to the curve, with algebraic criteria to certify pseudo-generic position and correct lifting at apparent singularities, ensuring exact results for all inputs.

ABSTRACT

An algorithm is presented for the computation of the topology of a non-reduced space curve defined as the intersection of two implicit algebraic surfaces. It computes a Piecewise Linear Structure (PLS) isotopic to the original space curve. The algorithm is designed to provide the exact result for all inputs. It's a symbolic-numeric algorithm based on subresultant computation. Simple algebraic criteria are given to certify the output of the algorithm. The algorithm uses only one projection of the non-reduced space curve augmented with adjacency information around some "particular points" of the space curve. The algorithm is implemented with the Mathemagix Computer Algebra System (CAS) using the SYNAPS library as a backend.

Motivation & Objective

• To address the lack of certified algorithms for computing the topology of non-reduced space curves in algebraic geometry.
• To provide a symbolic-numeric method that guarantees exact results for all inputs, avoiding numerical instability.
• To reduce computational cost by using only one projection, unlike prior methods requiring two.
• To introduce and certify pseudo-generic position for space curves to enable correct lifting of topology from the projected curve.
• To develop a robust algorithm that handles apparent singularities and distinguishes branches using slope and limit analysis.

Proposed method

• The algorithm projects the non-reduced space curve onto the (x,y)-plane using subresultant sequences to compute the implicit equation of the projection.
• It uses a rational parametrization derived from subresultants to lift the topology from the plane curve back to 3D space without numerical errors.
• Pseudo-generic position is certified via algebraic criteria ensuring that almost every (x,y) point has a unique z-preimage, except at singularities.
• At apparent singularities, the algorithm distinguishes branches using the slopes of the projected curve’s components and limit analysis of z-coordinates via Taylor expansions.
• The algorithm applies Sturm-Habicht sequences and subresultant computations to detect and classify singularities and to certify the correctness of the topology.
• The implementation uses the Mathemagix CAS with the SYNAPS library, and the output is a piecewise linear structure isotopic to the original curve.

Experimental results

Research questions

• RQ1 How can the topology of a non-reduced space curve defined by two implicit surfaces be computed exactly and certified?
• RQ2 Can a single projection suffice to reconstruct the full 3D topology of a space curve, including at singularities?
• RQ3 What algebraic criteria can certify that a space curve is in pseudo-generic position to ensure correct lifting from the projection?
• RQ4 How can apparent singularities—where multiple branches meet in projection but are distinct in 3D—be correctly resolved?
• RQ5 What is the computational cost and accuracy of a symbolic-numeric algorithm that avoids multiple projections and numerical approximations?

Key findings

• The algorithm computes a piecewise linear structure isotopic to the original non-reduced space curve using only one projection, achieving exact results.
• The method is certified via algebraic criteria for pseudo-generic position, ensuring correctness without reliance on numerical precision thresholds.
• Apparent singularities are resolved by analyzing tangent slopes and limits of z-coordinates via Taylor expansions, enabling correct branch connection.
• The implementation runs in 0.032 to 2.875 seconds on a 2GHz CPU with 1GB RAM, demonstrating practical efficiency on diverse curves.
• The algorithm avoids the need for a second projection, reducing computational cost compared to prior methods that used two projections.
• The use of rational parametrization from subresultants enables fast and numerically stable lifting of the projected topology to 3D space.