On the Bit Complexity of Solving Bilinear Polynomial Systems

Ioannis Z. Emiris, Angelos Mantzaflaris, Elias Tsigaridas
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16  
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
more » ... e 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 OB(n 4 D 4 + n 2 D 4 τ ), where n is the number of variables, D equals the bilinear Bézout 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. √ 3 3 ; −1, 2 ∓ √ 3) plus the root (x0, x1; y0, y1, y2) = (0, 1; 0, −1, 1) at infinity.
doi:10.1145/2930889.2930919 dblp:conf/issac/EmirisMT16 fatcat:ou73donorbhvfbwes7dzlrxaoy