# Probability that a Random Monic Polynomial Has Few Real Zeros

In the paper https://arxiv.org/pdf/math/0006113.pdf, it is shown that the probability that a random polynomial $$a_0 + a_1x + \cdots + a_n x^n$$ has $$o(\log n / \log\log n)$$ real zeros is $$n^{-b + o(1)}$$ as $$n \to \infty$$, where the coefficients $$a_i$$ are independently and identically distributed with zero mean and unit variance and where $$b > 0$$ is some absolute constant (see Theorem 1.2 on page 2).

Question: Are there any analogous results in the literature for monic polynomials? I.e., are there estimates for the probability that a random monic polynomial $$a_0 + a_1x + \cdots + a_{n-1} x^{n-1} + x^n$$ has few real zeros?

What I know: I am not sure that one can simply deduce the desired estimate in the case of monic polynomials from the case of non-monic polynomials. However, it seems possible that the method used in the aforementioned paper can be modified to give the desired estimate in the monic case, so I'm wondering if this has already been done. Also, if $$f(x_0) = 0$$ with $$x_0 \neq 0$$ and $$F(x,y)$$ is the homogenization of $$f(x)$$, then $$F(x_0,1) = 0$$, implying that $$g(y) := F(1,y)$$ satisfies $$g(1/x_0) = 0$$. Thus, the question above is related to the following question: are there estimates for the probability that a random polynomial of the form $$1 + a_1x + \cdots + a_nx^n$$ has few real zeros?