This work explores a method that reduces the design of evaluation strategies for geometric predicates to the computation of polynomial invariants of a group action. We apply it to the classical problem of counting line transversals to lines in P and capture polynomials previously obtained by more pedestrian approaches.

The enumeration of pseudo-intents is a long-standing problem in which the order plays a major role. In this paper, we present new algorithms that enumerate pseudo-intents in orders that do not necessarily respect the inclusion relation. We show that, confirming established results, this enumeration is equivalent to computing minimal transversals in hypergraphs a bounded number of times.

The enumeration of pseudo-intents is a long-standing problem in which the order plays a major role. In this paper, we present new algorithms that enumerate pseudo-intents in orders that do not necessarily respect the inclusion relation. We show that, confirming established results, this enumeration is equivalent to computing minimal transversals in hypergraphs a bounded number of times.

Let F = {X1, . . . , Xn} be a family of disjoint compact convex sets in R. An oriented straight line ` that intersects every Xi is called a (line) transversal to F . A transversal naturally induces an ordering of the elements of F , this ordering, together with its reverse (which is induced by the same line with opposite orientation), is called a geometric permutation. Geometric transversal the...

For each positive integer k, we consider the set A k of all ordered pairs [a, b] such that in every k-graph with n vertices and m edges some set of at most am + bn vertices meets all the edges. We show that each A k with k > 2 has infinitely many extreme points and conjecture that, for every positive e, it has only finitely many extreme points [a, b] with a _> e. With the extreme points ordered...

Let L(m;n) be the largest integer such that, if each symbol in an m n rectangle occurs at most L(m;n) times, then the array must have a transversal. We improve the lower bound to L(m;n) b m+1) 1 m 1 c, form > 1. Then we show that sporadically L(m;n) < bmn 1 m 1 c in the range m n m 3m + 3. De ne n0(m) to be the smallest integer z such that if n z then L(m;n) = bmn 1 m 1 c. We improve n0(m) from...