Computing a rational in between

Abstract

We are interested in the following problem: Given two (distinct) real algebraic numbers in isolating interval representation, that is an isolating interval with rational endpoints and a square free polynomial with integer coefficients, can we compute a number between them as a rational function of the coefficients of the polynomials that define these two numbers? Assume that the order of the two numbers is known (we will remove this assumption in the sequel). If we are given intervals that contain the real algebraic numbers and a procedure to refine them, we can solve our problem as follows: We refine the intervals until they become disjoint, this will happen eventually since we assume that the algebraic numbers are not equal, and then we compute a rational between the intervals, which separates the algebraic numbers. However, this iterative approach depends on separation bounds, e.g. [7]. We present a direct method, which is applicable when we allow in addition to compute the floor of a polynomial expression that involves real algebraic numbers. The problem arises when we wish to compute rational numbers that isolate the roots of an integer polynomial of small degree, say ≤ 5 [2]. Also in geometry, in order to analyse the intersection of two quadrics P and Q [6], one needs to determine the real roots of the polynomial det( P + xQ ) = 0, their multiplicities and a value in between each of these roots. Another motivation comes from the arrangement of quadrics [4] In this case a rational is needed separating two real roots of two polynomials with real algebraic numbers as coefficients. The real roots of such polynomials can be expressed as real algerbaic numbers and so we face the problem of computing a rational separating two real algebraic numbers.