Bit complexity for critical point computation in smooth and compact real hypersurfaces

Consider the polynomial mapping defined by the projection to the first coordinate on a real, smooth and compact hypersurface. The critical points of this mapping in generic coordinates have several applications in real algebraic geometry. We provide bit complexity estimates for computing them. Generic coordinates are obtained by applying a randomly chosen linear change of variables to the polynomial defining the hypersurface. The coordinates are sufficiently generic when the Jacobian matrix of the system under study has full rank at the critical points and when the number of critical points is finite. We have proven a new quantitative extension of Thom's weak transversality theorem [1]. By applying this extension, we are able to choose sufficiently generic changes of variables with arbitrarily high probability.