When n =1, then probability = 1 (one zero has to be real )
When n=2, then probability = 1/2 ( two zeros are either both real or both imaginary )
When n= 3, then probability = 1/2 ( either all zeros are real or one zero is real and two are imaginary)
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The answer depends on the joint distribution of the coefficients. If the coefficients are independent and the distribution on any single coefficient is symmetric around zero then the probability of having two real roots is greater than zero. The reason is as follows. If the polynomial is ax^2+bx+c then there are two real roots if b^2-4ac>0. The distribution of -4ac is symmetric around zero, so that the median of -4ac is zero. The term b^2 is greater than or equal to zero so the median of b^2-4ac is positive (except if b=0 almost surely).
Dear Peter,
1) What means the coefficients of a polynomial are independent?
2) The distribution on any single coefficient is symmetric around zero?
Please.
By the Fundamental Theorem of Algebra, any (monic) polynomial is a product (z-a1)(z-a2)....(z-an) with ai complex numbers.Now the probability of any ai being real is zero,since the real line has measure zero in the complex plane. So the probability of all roots being real is zero.
@ Srinivas Rau your argument is correct for polynomials where the coefficients are complex numbers, but the question was what happens for real polynomials.
@ Hanifa Zekraoui Before we can calculate any probabilities we have to have a probability distribution on the coefficients, i.e. we have to consider the coefficients as random variables. If the polynomial is ax^2+bx+c we should consider a,b, and c as random variables. One condition is that a, b, and c are independent random vaariables. Another condition is that each of the random variables has a symmetric distribution. For a it means that the distribution of a is the same as the distribution of -a. One such model is that a, b, and c are independent standard Gaussian random variables.
An important model with dependent coefficients when the polynomial is the characteristic polynomial of a Hermitean matrix. In this case all the roots automatically real.
Well,for a fixed degree n,the subset of polynomials with real coefficients is negligible; of these there are polynomials with only complex roots ,like x^2+1. On the other hand,every odd degree real polynomial has a real root.Perhaps one can pose the question for the subclass of odd degree real polynomials.
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