If we randomly pick a n-degree polynomial, is the probability of getting complex roots higher than the probability of getting all real roots?

Question

Intuitively, I feel the probability of getting all roots of a polynomial to be real to be less. But is there a proof for this statement?

Answer 1

A Google search for "distribution of number of real roots in a polynomial" came up with this as the first hit:

https://web.williams.edu/Mathematics/sjmiller/public_html/ntprob12/handouts/polyzeros/Fairley_NumbRealRootsRandPolySmallDeg.pdf

Here is the title and abstract:

THE NUMBER OF REAL ROOTS OF RANDOM POLYNOMIALS OF SMALL DEGREE*

By WILLIAM B. PAIRLEY Kennedy School of Government, Harvard University

SUMMARY. Random polynomials with random coefficients have been studied by a number of authors, including Kac (1943, 1949), who showed that the average number of real roots of polynomials of degree n is asymptotically $(2/\pi) \log n$. The present paper investigates the average number of real roots for polynomials of small degree and coefficients that are 1 or - 1 with equal probability. Log-like behavior of the average for small n is shown by finding exact distributions of the numbers of real roots for n between 1 and 10 and by sampling the large but finite populations for n between 10 and 50.

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Since the number of real roots of a polynomial of degree $n$ is about $(2/\pi) \log n$, the vast majority of roots are complex.

Here are the references:

Bloch, A. and Polya, G. (1932) : On the roots of certain algebraic equations. Proc. London Math. Soc, II, 33, 102-114.

Erdos, P. and Offord, A. (1956) : On the number of real roots of a random algebraic equation. Proc. London Math. Soc, 6, 139-160.

Ibragimov, I. and Maslova, N. (1971) : On the expected number of real zeros of random polynomials I. Coefficients with zero means. Theory of Probability and its Applications, XVI, No. 2.

Kac, M. (1943) : On the average number of real roots of a random algebraic equation. Bulletin American Math. Soc, 49, 314-320.

-(1949) : On the average number of real roots of a random algebraic equation (II). Proc. London Math. Soc, 50, 390-408.

Lilley, F. (1967) : Zeros of a Polynomial. General electric information series, No. 65SD351.

Littlewood, J. and Offord, A. (1939) : On the number of real roots of a random algebraic equation II. Proc Cambridge Phil. Soc, 35, 133-148.

Stevens, D. (1965) : The average number of real zeros of a random polynomial, doctoral dissertation, New York University, Department of Mathematics.