Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets

TL;DR

This paper presents polynomial-time algorithms for computing the generalized Euler-Poincaré characteristic of symmetric semi-algebraic sets, significantly improving over the known singly exponential algorithms for non-symmetric cases.

Contribution

It introduces efficient algorithms with polynomial complexity for symmetric semi-algebraic sets, contrasting with the exponential complexity in non-symmetric cases.

Findings

Algorithms have polynomial complexity in the number of polynomials and dimension.

Complexity is polynomial when polynomial degrees are bounded.

Symmetric case algorithms outperform non-symmetric algorithms in complexity.

Abstract

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincar\'e characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.