In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Monte Carlo probabilistic algorithm which solves this problem, under some regularity assumptions on the input, by taking advantage of the symmetry invariance property. The complexity of our algorithm is polynomial in $d^s, {{n+d} \choose d}$, and ${{n} \choose {s+1}}$, where $n$ is the number of variables and $d$ is the maximal degree of $s$ input polynomials defining the real algebraic set under study. In particular, this complexity is polynomial in $n$ when $d$ and $s$ are fixed and is equal to $n^{O(1)}2^n$ when $d=n$.

翻译：本文考虑具有实系数的多项式方程组实解存在性判定问题，该方程组在对称群作用下具有不变性。我们构建并分析了一种蒙特卡洛概率算法，通过利用对称不变性性质，在输入满足正则性假设的前提下解决该问题。该算法的复杂度为 $d^s, {{n+d} \choose d}$ 及 ${{n} \choose {s+1}}$ 的多项式形式，其中 $n$ 表示变量个数，$d$ 为定义所研究实代数集的 $s$ 个输入多项式的最高次数。特别地，当 $d$ 和 $s$ 固定时该复杂度为 $n$ 的多项式，而当 $d=n$ 时复杂度为 $n^{O(1)}2^n$。