Dennis S. Arnon

Cylindrical algebraic decomposition I: the basic algorithm

Given a set of r-variate integral polynomials, a cylindrical algebraic decomposition (cad) of euclidean r-space E r partitions E r into connected subsets compatible with the zeros of the polynomials. By “compatible with the zeros of the polynomials” we mean that on each subset of E r , each of the polynomials either vanishes everywhere or nowhere. For example, consider the bivariate polynomial

A polynomial-time algorithm for the topological type of a real algebraic curve

{y^4} - 2{y^3} + {y^2} - 3{x^2}y + 2{x^4}.

A Cluster-Based Cylindrical Algebraic Decomposition Algorithm

In Part I of the present paper we defined cylindrical algebraic decompositions (cad’s), and described an algorithm for cad construction. In Part II we give an algorithm that provides information about the topological structure of a cad of the plane. Informally, two disjoint cells in E r , r ≥ 1, are adjacent if they touch each other; formally, they are adjacent if their union is connected. In a picture of a cad, eg Fig. 2 of Part I, it is obvious to the eye which pairs of cells are adjacent. However, the cad algorithm of Part I does not actually produce this information, nor did the original version of that algorithm in (Collins 1975).

An Adjacency Algorithm for Cylindrical Algebraic Decompositions of Three-Dimensional Space

It was proved over a century ago that an algebraic curve C in the real projective plane, of degree n, has at most (n-1)(n-2)2+1 connected components. If C is nonslngular, then each of its commponents is a topological circle. A circle in the projectlve plane either separates it into a disk (the interior of the circle) and a Mobius band (the circles exterior), or does not separate it. In the former case, the circle is an oval. If C is nonsingular, then all its components are ovals if n is even, and all except one are ovals if n is odd. An oval is included in another if it lies in the others interior. The topological type of (a nonsingular) C is completely determined by (1) the parity of n, (2) how many ovals it has, and (3) the partial ordering of its ovals by inclusion. We present an algorithm which, given a homogeneous polinomial f(x,y,z) of degree n with integer coefficients, checks whether tlte curve defined hy f = 0 is nonsingular and if so, computes its topological type. The algorithms maximum computing time is O(n^2^7L(d)^3), where d is the sum of the absolute values of the integer coofficients of f, and L(d) is the length of d.

CaminoReal: an interactive mathematical notebook

Abstract Geometric reasoning is concerned with (geometric) objects that often are definable by formulae of the language of the first-order theory of the real numbers. Certain problems that arise in geometric reasoning can be cast into one of the following forms: query problem—does a certain collection of objects possess a certain (first-order) property; constraint problem—given a quantified formula defining an object, find an equivalent definition by a quantifier-free formula; display problem—describe an object, e.g. determine its dimension or specify its topology. In the last fifteen years, feasible algorithms for the exact solution of these problems have been discovered, implemented, and used to solve nontrivial problems. We give examples of problems that fall within the scope of these methods, provide a tutorial introduction to the principal algorithms currently in use, and describe the solution of the sample problems using those algorithms.

A cluster-based cylindrical algebraic decomposition algorithm

A basic collection of literature relatlng to algorithmic quantifier elimination for real closed fields is assembled.

On mechanical quantifier elimination for elementary algebra and geometry

Given a set A of r-variate integral polynomials, an A-invariant cylindrical algebraic decomposition (cad) of euclidean r-space Er is a certain partition of Er into connected subsets, such that each subset is an i-cell for some i, 0≤i≤r (an i-cell is a homeomorph of Ei), and such that each element of A is sign-invariant (i.e. either negative, vanishing, or positive) on each cell. A cluster of a cad D is a collection of cells of D whose union is connected. We give a new cad construction algorithm which, as it extends a cad of Ei−1 to a cad of Ei (2≤i≤r), partitions the cells of the Ei−1 cad into clusters, and performs various computations only once for each cluster, rather than once for each cell as previous cad algorithms do. Preliminary experiments suggest that this new algorithm can be significantly more efficient in practice than previous cad algorithms. The clusters, which are part of the new algorithms output, can be chosen to have useful mathematical properties. For example, if r≤3, each cluster (i.e. the union of its cells) can be a maximal connected A-invariant subset of Er.