Real Root Finding for Equivariant Semi-algebraic Systems

Let R be a real closed field. We consider basic semi-algebraic sets defined by n -variate equations/inequalities of s symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2d < n. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d-1 distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by s polynomials of degree d in time (sn)O(d). This improves the state-of-the-art which is exponential in n . When the variables x1, łdots, xn are quantified and the coefficients of the input system depend on parameters y1, łdots, yt, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time (sn)O(dt).