Rational approximation to real points on conics

Rational quadratic approximation to real algebraic curves

An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic B-spline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The f...

Rational Approximation on the Positive Real Axis

Introduction Rational Chebyshev approximation to reciprocals of certain entire functions by reciprocals of polynomials on the positive real axis has recently attracted the attention of many mathematicians . By developing certain new methods of approach we successfully attacked ([3]-[6]) some of the related problems . This paper is a continuation of our earlier papers ([3]-[6] ) . The results of...

The Number of Rational Points on Conics Cp,k : x2 − ky2 = 1 over Finite Fields Fp

Let p be a prime number, Fp be a finite field, and let k ∈ Fp. In this paper, we consider the number of rational points on conics Cp,k : x − ky = 1 over Fp. We proved that the order of Cp,k over Fp is p− 1 if k is a quadratic residue mod p and is p+1 if k is not a quadratic residue mod p. Later we derive some results concerning the sums ∑ C [x] p,k(Fp) and ∑ C [y] p,k(Fp), the sum of x− and y−c...

Eecient Solution of Rational Conics

Rational Points on Pencils of Conics and Quadrics with Many Degenerate Fibres

For any pencil of conics or higher-dimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics over Q, which is a consequence of recent advances in additive combinatorics.