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The study of random polynomials has a long and rich history. This paperstudies 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 takenas the increments $X(k+1) - X(k)$, $k\in \mathbb{N}$, of a fractional Brownianmotion $X$ of Hurst index $0< H < 1$. This reduces to the classical setting ofindependent coefficients for $H = 1/2$. We obtain that the average number ofthe 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 Kacin 1943). Unexpectedly, the parameter $H$ affects only the number of positivezeros, and the number of real zeros of the polynomials corresponding tofractional Brownian motions of indexes $H$ and $1-H$ are essentially the same.The limit case $H = 0$ presents some particularities: the average number ofpositive zeros converges to a constant. These results shed some light on thenature of fractional Brownian motion on the one hand and on the behaviour ofreal zeros of random polynomials of dependent coefficients on the other hand.

Average number of real roots of random polynomials defined by the increments of fractional Brownian motion