In this paper we study the distribution of the real algebraic numbers. Given an interval I, a positive integer n and Q > 1, define the counting function Φ n (Q; I) to be the number of algebraic numbers in I of degree n and height ≤ Q.The distribution function is defined to be the limit (as Q → ∞) of Φ n (Q; I x ) divided by the total number of real algebraic numbers of degree n and height ≤ Q. We prove that the distribution function exists and is continuously differentiable. We also give an explicit formula for its derivative (to be referred to as the distribution density) and establish an asymptotic formula for Φ n (Q; I) with upper and lower estimates for the error term in the asymptotic. These estimates are shown to be exact for n ≥ 3. One consequence of the main theorem is the fact that the distribution of real algebraic numbers of degree n ≥ 2 is non-uniform.

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Given a polynomial q(z) := a 0 + a 1 z + · · · + anz n and a vector of positive weights w = (w 0 , w 1 , . . . , wn), define the w-weighted lp-norm of q as lp,w[q] := n k=02010 Mathematics Subject Classification. Primary, 11N45; secondary, 11C08, 60G55, 30C15, 26C10.

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]