  The study of random polynomials has a long and rich history. This paper
studies random algebraic polynomials $P_n(x) = a_0 + a_1 x + \ldots + a_{n-1}
x^{n-1}$ where the coefficients $(a_k)$ are correlated random variables taken
as the increments $X(k+1) - X(k)$, $k\in \mathbb{N}$, of a fractional Brownian
motion $X$ of Hurst index $0< H < 1$. This reduces to the classical setting of
independent coefficients for $H = 1/2$. We obtain that the average number of
the real zeros of $P_n(x)$ is~$\sim K_H \log n$, for large $n$, where $K_H = (1
+ 2 \sqrt{H(1-H)})/\pi$ (a generalisation of a classical result obtained by Kac
in 1943). Unexpectedly, the parameter $H$ affects only the number of positive
zeros, and the number of real zeros of the polynomials corresponding to
fractional Brownian motions of indexes $H$ and $1-H$ are essentially the same.
The limit case $H = 0$ presents some particularities: the average number of
positive zeros converges to a constant. These results shed some light on the
nature of fractional Brownian motion on the one hand and on the behaviour of
real zeros of random polynomials of dependent coefficients on the other hand.
