Re: probability of real factorization.

In article <1133353263.758335.202840@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<pauldepstein@xxxxxxx> wrote:
>Let n be a fixed positive integer.
>
>Let N also be a positive integer.
>
>Randomly select a real monic polynomial in Z[x] of degree n such that
>all coefficients have absolute value less than N. (Select such that
>all eligible polynomials have equal probability of being chosen.)
>
>As N tends to infinity, what is the limit of the probability, p_n,
>that the selected polynomial factors completely over the reals.
>
>For n = 2, p_n = 1 I think.

For n = 3, the discriminant of x^3 + a x^2 + b x + c is
D(a,b,c) = -27 c^2 + 18 a b c + a^2 b^2 - 4 a^3 c - 4 b^3.
The cubic has three real roots iff D(a,b,c) >= 0.
Note that if |a|, |b|, |c| <= N,
D(a,b,c) = a^2 (b^2 - 4 a c) + O(N^3)
so as N -> infty, the probability of three real roots should
approach the probability that b^2 - 4 a c > 0, which is
41/72 + ln(2)/12 (see Rob Johnson's posting).

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


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