Suppose we have a function $f$ in p-variables $x_1,...,x_p$ which has the form $f(x_1,...,x_p) = f_1(x_1)+...+f_p(x_p)$. Assume further that each function $f_p$ is a piecewise linear spline. What I would like to know is if there is a simple algorithm to find the points say $\bar{x}_1,...,\bar{x}_p$ where $f(\bar{x}_1,...,\bar{x}_p) = 0$ holds. I could imagine that we could go through all the intervalls where the spline has a break - for each direction. I suppose however that such an algorithm grows exponentially with dimension $p$.
Is there any faster way to do this ??