On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

Abstract

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas.The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials.

Sparse Rational Univariate Representation

“…We can also opt for a complexity of OB(E ω M0τ ), if we use the techniques for fast determinant computation [41], where ω is the exponent of matrix multiplication. We also refer to [9] for a recent result on the bit complexity of solving polynomial systems that exploits this computation. For sparse resultant matrix constructions, E stands for the number of lattice points of the polytope that is the Minkowski sum of the Newton polytopes of the input polynomials, e.g.…”

“…Taking into account the height bounds of SRUR, the bit complexity for computing the R is OB(n 2 M 4 τ ). Alternatively, we can use [41], see also [9], to compute the determinant in OB((nM) ω+1 τ ).…”

“…(9). To obtain the bounds, we consider the worst case scenario where all the coefficients are perturbed.…”

“…For each such constructible set the resultant of the system R1(t, u) is a homogeneous polynomial in the coefficients of the input polynomials and its terms are as in Eq. (9).…”

Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

“…Then Res(p(s, t) · X, f (s), g(s, t)) are the real extraneous factors (possibly with powers) associated to the base points determined by {f (s), g(s, t)}. Some techniques for solving for this zerodimensional variety can be found in [1,9,19].…”