On the index of special perfect polynomials

We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that $\omega(d) \geq 9$, and $\deg(d) \geq 258$, where $d := \gcd(Q^2,\sigma(Q^2))$ is the index of the special perfect polynomial $A := p_1^2 Q^2$, in which $p_1$ is irreducible and has minimal degree. This means that $ \sigma(A)=A$ in the polynomial ring ${\mathbb{F}}_2[x]$. The function $\sigma$ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of $135$ is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165$-$174].