For a set \(A:=\{ a_1,\ldots ,a_{n+2}\}\subset \mathbb{Z}^n\) of cardinality \(n+2\), with all coordinates of \(a_j\) having absolute value at most \(d\) and the \(a_j\) not all lying in the same affine hyperplane, one sets \(x^{a_j}:=x_1^{a_{1,j}}\cdots x_n^{a_{n,j}}\) and one considers the \(n\) functions \(f_i:=\sum _jc_{i,j}x^{a_j}\in \mathbb{Z}[x_1^{\pm 1},\ldots ,x_n^{\pm 1}]\).
Suppose that \(F:=(f_1,\ldots ,f_n)\) is an \(n\times n\) polynomial system with generic integer coefficients at most \(H\) in absolute value, and \(A\) the union of the sets of exponent vectors of the \(f_i\). The author gives the first algorithm that, for any fixed \(n\), counts exactly the number of real roots of \(F\) in time polynomial in log(d\(H\)). He also discusses a number-theoretic hypothesis that would imply a further speed-up to time polynomial in \(n\) as well.