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

Abstract

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system, under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set is finite. The algorithm is probabilistic and a probability analysis is provided.Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem.

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

“…This relies on precise bounds on the coefficient sizes of the polynomials P and the Q j appearing in the Kronecker representation in this situation. Such bounds have been provided recently by Safey El Din and Schost [61], extending earlier results of Schost [63] thanks to new height bounds by D'Andrea et al [20]. They allow us to determine the complexity of rigorously deciding several properties of the solutions to the original polynomial system needed in Steps 2 and 3 of the Algorithms Minimal Critical Points in the Combinatorial Case and Minimal Critical Points in the Non-Combinatorial Case.…”

“…Our work on this system relies on ideas by a variety of authors [27,28,63,40] on the use of the Kronecker representation in complex or real geometry, which go far beyond the simple systems we consider here. More precisely, we make use of the recent work of Safey El Din and Schost [61], who take into account multi-homogeneity and provide estimates on the height of the representations and the bit complexities of their algorithms. Note that as this work reached completion, a new preprint by van der Hoeven and Lecerf [66] appeared that points to the possibility of improving further the exponent of d n in our results, while retaining the same approach.…”

“…Next, in Section 4, we introduce the Kronecker representation. The results of Safey El Din and Schost [61] that we need are recalled. They are used to analyze the cost of several other operations on solutions of polynomial systems.…”

“…. , f n ) is a polynomial system where f j has multi-degree at most d j ∈ N m , and the block of variables Z j contains n j elements, Safey El Din and Schost [61] give an upper bound C n (d) on the degrees of the polynomials appearing in a Kronecker representation of the non-singular solutions of f , where d = (d 1 , . .…”

Sparse Rational Univariate Representation

“…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.…”