Counting real algebraic numbers with bounded derivative of minimal polynomial

Real algebraic numbers and polynomial systems of small degree

Based on precomputed Sturm-Habicht sequences, discriminants and invariants, we classify, isolate with rational points, and compare the real roots of polynomials of degree up to 4. In particular, we express all isolating points as rational functions of the input polynomial coefficients. Although the roots are algebraic numbers and can be expressed by radicals, such representation involves some r...

Counting Algebraic Numbers with Large Height Ii

We count algebraic numbers of fixed degree over a fixed algebraic number field. When the heights of the algebraic numbers are bounded above by a large parameter H, we obtain asymptotic estimates for their cardinality as H → ∞.

extensions of some polynomial inequalities to the polar derivative

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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 ...

On approximation of real numbers by real algebraic numbers