On the number of good approximations of algebraic numbers by algebraic numbers of bounded degree

Approximation of complex algebraic numbers by algebraic numbers of bounded degree

To measure how well a given complex number ξ can be approximated by algebraic numbers of degree at most n one may use the quantities w n (ξ) and w * n (ξ) introduced by Mahler and Koksma, respectively. The values of w n (ξ) and w * n (ξ) have been computed for real algebraic numbers ξ, but up to now not for complex, non-real algebraic numbers ξ. In this paper we compute w n (ξ), w * n (ξ) for a...

On the Approximation to Algebraic Numbers by Algebraic Numbers

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...

Computing algebraic numbers of bounded height

We describe an algorithm which, given a number field K and a bound B, finds all the elements of K having relative height at most B. Two lists of numbers are computed: one consisting of elements x ∈ K for which it is known with certainty that HK(x) ≤ B, and one containing elements x such that |HK(x)− B| < θ for a tolerance θ chosen by the user. We show that every element of K whose height is at ...

Comparing Real Algebraic Numbers of Small Degree

We study polynomials of degree up to 4 over the rationals or a computable real subfield. Our motivation comes from the need to evaluate predicates in nonlinear computational geometry efficiently and exactly. We show a new method to compare real algebraic numbers by precomputing generalized Sturm sequences, thus avoiding iterative methods; the method, moreover handles all degenerate cases. Our f...

On approximation of real numbers by real algebraic numbers