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

## TL;DR

This paper improves the bit complexity analysis of an algorithm for finding at least one point per connected component of smooth real algebraic sets, achieving exponential speedup by leveraging the multi-affine structure of polynomial systems.

## Contribution

It provides a refined complexity analysis that significantly enhances the efficiency of the critical point method for real algebraic sets, with better output size estimates.

## Key findings

- Exponential speedup in complexity compared to previous methods
- Improved bit-size estimates for output polynomials
- Enhanced utilization of multi-affine structure in polynomial systems

## 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.

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Source: https://tomesphere.com/paper/2508.20607