Faster real root decision algorithm for symmetric polynomials

Abstract: 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 $d=n$.