On the Bit Complexity of Solving Bilinear Polynomial Systems

Overview & Synopsis

We bound the Boolean complexity of computing isolating hyperboxes for all complex roots of systems of bilinear polynomials. The resultant of such systems admits a family of determinantal Sylvester-type formulas, which we make explicit by means of homological complexes. The computation of the determinant of the resultant matrix is a bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector products, corresponding to multivariate polynomial multiplication. For zero-dimensional systems, we arrive at a primitive element and a rational univariate representation of the roots. The overall bit complexity of our probabilistic algorithm is O_B(n^4 D^4 + n^2 D^4 τ), where n is the number of variables, D equals the bilinear Bezout bound, and τ is the maximum coefficient bitsize. Finally, a careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making our algorithms and complexity analysis applicable to the general case.