On the Bit Complexity of Solving Bilinear Polynomial Systems

Abstract

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 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.

Sparse Rational Univariate Representation

“…Usually, c · R1 = det(M ), where c is a constant; for example this is the case for the classical Macaulay resultant, the sparse resultant, etc. However, there are many cases where we have an exact determinantal representation for the resultant, for example the bilinear case [23], or the tensor-product polynomial systems that we consider next.…”

“…To compute SRUR we follow [23] that generalizes the method of Canny [10]. The general idea is to use resultant computations to obtain a representation of the roots of (Σ) using the primitive element, and then to convert this representation to the one of Eq.…”

“…However, there are cases where g0 is not linear. For example this is the case when we solve systems of bilinear polynomials using resultant matrices [23] or tensor-product polynomial systems, see Sec. 3.…”

“…We can compute the determinant of such matrices in O(nM 2 ) arithmetic operations by exploiting the algorithms for multivariate polynomial multiplication using Wiedemann's method e.g. [13,23]. Taking into account the height bounds of SRUR, the bit complexity for computing the R is OB(n 2 M 4 τ ).…”

“…This representation is related to the arithmetic Nullstellensätz [18,30,39,40], see also [33] for the most recent approach, and the separation bounds of the polynomial systems [21]. There are also dedicated estimates for the special cases of bivariate [8,34], bilinear [23], and multi-homogeneous [36] polynomial systems. Our references represent only the tip of the iceberg of the existing ones on the subject.…”