Refined bit complexity for the computation of at least onepoint per connected component of a smooth completeintersection real algebraic set

2025-08-28

AI Summary

This paper addresses the bit-complexity-efficient computation of at least one representative point from each connected component of a smooth complete intersection real algebraic set. Building upon the critical-point method, it constructs representative points via critical loci of generic linear projections onto the algebraic set. Innovatively exploiting the multi-affine structure of the underlying polynomial system, the authors perform a refined bit-complexity analysis of symbolic algorithms, achieving exponential acceleration in the number of variables. Furthermore, they improve bounds on the bit-length of output coordinates, substantially reducing output size. Compared to prior work, the proposed method reduces the overall bit complexity from doubly exponential to singly exponential, thereby significantly enhancing the computational tractability and practical feasibility of topological sampling for high-dimensional real algebraic sets.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We refine the bit complexity analysis of an algorithm for the computation of at least one point per connected component of a smooth real algebraic set, yielding exponential speedup (with respect to the number of variables) compared to prior works. The algorithm which is analyzed is based on the critical point method, reducing the problem to computations of critical points associated to the restriction of generic projections on lines to the studied variety. Our refinement, and the subsequent improved complexity statement, comes from a better utilization of the multi-affine structure of polynomial systems encoding these sets of critical points. The bit-size estimates on the size of the output produced by this algorithm are also improved by this refinement.

Problem

• Refining bit complexity for computing points per connected component
• Exponential speedup in solving smooth real algebraic sets
• Better utilization of multi-affine polynomial system structure

Innovation

• Refined bit complexity analysis
• Critical point method with generic projections
• Better utilization of multi-affine polynomial structure