The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials Pn(x)=a0+a1x+⋯+an-1xn-1\documentclass[12pt]{minimal}
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\begin{document}$$P_n(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1}$$\end{document} where the coefficients (ak)\documentclass[12pt]{minimal}
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\begin{document}$$(a_k)$$\end{document} are correlated random variables taken as the increments X(k+1)-X(k)\documentclass[12pt]{minimal}
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\begin{document}$$X(k+1) - X(k)$$\end{document}, k∈N\documentclass[12pt]{minimal}
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\begin{document}$$k\in \mathbb {N}$$\end{document}, of a fractional Brownian motion X of Hurst index 0<H<1\documentclass[12pt]{minimal}
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\begin{document}$$0< H < 1$$\end{document}. This reduces to the classical setting of independent coefficients for H=1/2\documentclass[12pt]{minimal}
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\begin{document}$$H = 1/2$$\end{document}. We obtain that the average number of the real zeros of Pn(x)\documentclass[12pt]{minimal}
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\begin{document}$$P_n(x)$$\end{document} is ∼KHlogn\documentclass[12pt]{minimal}
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\begin{document}$$\sim K_H \log n$$\end{document}, for large n, where KH=(1+2H(1-H))/π\documentclass[12pt]{minimal}
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\begin{document}$$K_H = (1 + 2 \sqrt{H(1-H)})/\pi $$\end{document} [a generalisation of a classical result obtained by Kac (Bull Am Math Soc 49:314–320, 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\documentclass[12pt]{minimal}
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\begin{document}$$1-H$$\end{document} is essentially the same. The limit case H=0\documentclass[12pt]{minimal}
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\begin{document}$$H = 0$$\end{document} 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.
