Deformation Techniques for Sparse Systems

Abstract

Abstract. We exhibit a probabilistic symbolic algorithm for solving zerodimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is quadratic in the size of the combinatorial structure of the input system. This size is mainly represented by the mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration.

Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization

“…As already said, the techniques used in the algorithm are not new: we first solve the system modulo a prime, using a symbolic homotopy algorithm that adapts to the multi-homogeneous case an algorithm given by Jeronimo et al [26] for the sparse case; then, we use lifting techniques from [21,44], as well as techniques coming from [41,Section 4], to recover the output over Z. Taking into account our upper bound on the height of the output, this results in the first bound (that we are aware of) on the boolean cost of solving polynomial systems that involves their multi-homogeneous structure in such a manner.…”

“…, d i,m ). Closer to us are two algorithms from [21] and [26]. The geometric resolution algorithm of [21] solves our questions in time quadratic in a particular geometric degree associated to the input system; however, in general, this degree cannot be controlled in terms of the quantities C n (d) and C n (d ) used in our analysis (see for example those systems appearing in [25]); in addition, we are not aware of a probability analysis for it.…”

“…These results are for a substantial part not new. The algorithm can in particular be seen as a modification of that in [26]; we do however have to give a rather detailed presentation, for reasons explained in Subsection 3.1.…”

“…The algorithm of [26] deals with symbolic homotopies for sparse systems, with a running time that would be comparable to ours in the case of multi-homogeneous systems. However, that algorithm requires a base field of characteristic zero (whereas we will need it over a finite field), and the system f must be zero-dimensional (which is not the case for us); in addition, the last step of that algorithm, specialization at t = 1 (Section 6.2 in [26]) appears to overlook issues that we discuss below, inspired by [41,Section 4]. For these reasons, lacking another reference, we decided to include a self-contained proof dedicated to our multi-homogeneous situation.…”

On Sign Conditions Over Real Multivariate Polynomials

“…So, we first compute a geometric resolution associated with α (x) for each S B,e : after solving a linear system, the x-coordinates of points in S B,e turn to be defined by a square polynomial system in separated variables; then, the required computation can be achieved as in [27,Sect. 5 The whole complexity of this step is O(nD 2 log 2 (D) log log(D)).…”