Mixed-volume computation by dynamic lifting applied to polynomial system solving

J. Verschelde, K. Gatermann, R. Cools
1996 Discrete & Computational Geometry  
The aim of this paper is to present a flexible approach for the efficient computation of the mixed volume of a tuple of polytopes. In order to compute the mixed volume, a mixed subdivision of the tuple of polytopes is needed, which can be obtained by embedding the polytopes in a higher-dimensional space, i.e., by lifting them. Dynamic lifting is opposed to the static approach. This means that one considers one point at a time and only fixes the value of the lifting function when the point
more » ... influences the mixed volume. Conservative lifting functions have been developed for this purpose. This provides us with a deterministic manipulation of the lifting for computing mixed volumes, which rules out randomness conditions. Cost estimates for the algorithm are given. The implications of dynamic lifting on polyhedral homotopy methods for the solution of polynomial systems are investigated and applications are presented. Mixed-Volume Computation by Dynamic Lifting Applied to Polynomial System Solving 71 Dynamic Construction of Regular Triangulations This part describes the dynamic lifting algorithm applied to one polytope. Its structure is as follows. After some preliminary definitions, the basic version of the algorithm is sketched. The following subsections describe the key steps in this algorithm. Some cost estimates are given in the final subsection. Regular Triangulations For completeness we start with some well-known definitions [34] . We assume the points to be vectors in Euclidean space E", equipped with the standard inner product (-, .). Definition 2.1. Given a set A c E". The dimension of A equals d, denoted by dim(A) = d, if A contains at most d + 1 points Co, el ..... Cd such that e0 --cj, j = 1, 2 ..... d, are linearly independent, For a finite set A C E", the convex hull of A, denoted by conv(A), is the smallest convex set that contains A. The polytope P of A is defined by P = conv(A). A face of a polytope P is the intersection of hyperplanes which define half-spaces that contain the polytope entirely. The polytope itself is considered as a trivial face. All other faces, the empty set included, are proper faces. A vertex of a polytope is a face of dimension zero. A face of dimension k is called a k-face. Definition 2.2. Given a polytope P in n-dimensional space, with dim(P) = d. A facet 0 P of P is a face of P, with dim(0 P) = d-1.0 P is defined as the intersection of P with one hyperplane that defines a half-space which contains P entirely and is characterized by its inner normal y: 1. u y ~ OP, (x, y) = (y, y). 2. (-, y) attains its minimum at OP, i.e., (y, ~/) > (x, y), u e OP, u ~ OP. Since the functional (-, y) is constant on OP we denote (OP, y) := (x, y), for one x e 0 P. The facet itself is denoted by O r P = cony(0 r A), with 0 e A = {x E A I (x, y) = miny~a (y, y)}. Definition 2.3. The lower hull of a polytope P consists of all facets 0 r P with y, > 0. The following definitions are based on the definitions in [30] and in [35]. See Lecture 5 of [56] for a more detailed mathematical background. Definition 2.4. Given a set of points A C E n, a subdivision S of A consists of a collection of cells S = {C1, C2 ..... Cm}, with Ck C A, Vk, which satisfies: I, dim(Ck) = n. 2. conv(C/) N conv(Ck) is a proper face of both conv(Cl) and conv(Ck), I # k. 3. Ukm__l conv(Ck) = conv(A).
doi:10.1007/bf02711134 fatcat:wsbnzecc4jcbxkjgbssq7b6pjy