We investigate the one-parametric set G of projective subspaces that is generated by a set of rational curves in projective relation. The main theorem connects the algebraic degree δ of G, the number of degenerate subspaces inG and the dimension of the variety of all rational curves that can be used to generateG. It generalizes classical results and is related to recent investigations on projec...

In the study of farsighted coalitional behavior, a central role is played by the von Neumann-Morgenstern (1944) stable set and its modification that incorporates farsightedness. Such a modification was first proposed by Harsanyi (1974) and has recently been re-formulated by Ray and Vohra (2015). The farsighted stable set is based on a notion of indirect dominance in which an outcome can be domi...

Let E be an elliptic curve defined over Q and P ∈ E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the quadratic imaginary field k. Denote by ME,P the set of rational primes ` such that ` splits in k, E has good reduction at `, and P is a primitive point modulo `. Under the generalized Riemann hypothesis, we can determine the positivity of the de...

Let f(z) = z +az + bz + cz+ d ∈ Z[z] and let us consider a del Pezzo surface of degree one given by the equation Ef : x 2 − y − f(z) = 0. In this note we prove that if the set of rational points on the curve Ea, b : Y 2 = X + 135(2a − 15)X − 1350(5a + 2b − 26) is infinite, then the set of rational points on the surface Ef is dense in the Zariski topology.

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over Q. The focus is on practical aspects of this problem in the case that the genus of C is at least 2, and therefore the set of rat...

Current CAD systems can represent curves and surfaces only in rational B-spline (NURBS) form ( .Farin, 1994; .Hoschek and Lasser, 1993). On the other hand, certain curves and surfaces that arise in practical applications such as offsets of rational curves or surfaces are in general not rational and therefore need to be approximated. This motivated .Farouki and Sakkalis (1990) to introduce the s...

Let K be a number field and let C be a curve of genus g > 1 defined over K. In this dissertation we describe techniques for bounding the number of K-rational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpublished. We have tried to eliminate unnecessary restrictions, such as assumptions of good reducti...

If P is an algebraic point on a commutative group scheme A/K, then P is almost rational if no two non-trivial Galois conjugates σP , τP of P have sum equal to 2P . In this paper, we classify almost rational torsion points on semistable elliptic curves over Q. 1 Definitions and Results Let X be an algebraic curve of genus greater than one. Let J(X) be the Jacobian variety of X, and embed X in J(...

Given a quadratic polynomial q : Ĉ→ Ĉ and a representation G : Ĉ→ Ĉ of C2 ∗C5 in PSL(2,C) satisfying certain conditions, we will construct a 4 : 4 holomorphic correspondence on the sphere (given by a polynomial relation p(z,w)) that mates the two actions: The sphere will be partitioned into two completely invariant sets Ω and Λ. The set Λ consists of the disjoint union of two sets, Λ+ and Λ−, e...

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups as well as the set of all solutions of equations in free monoids with involution in the presence of rational constraints. This became possible due to the recently invented recompression technique of the second author. He ...