# Expected roots of polynomials with randomness in coefficients

I need some results on expected roots of polynomial with random coefficients. I searched for $$\textit{roots of polynomial with random coefficients}$$ online and found some papers, but they are dealing with more complicated problems than I need. I think my particular problem is more specific: suppose we have a, say, cubic polynomial equation $$p_1(x) = ax^3 + bx^2 + cx + d = 0$$ where $$a,b,c,d$$ are some deterministic real numbers.

Suppose that a polynomial with random coefficients (in my case, the highest degree coefficient is not random, not sure if this will make a difference or not) is $$p_2(x) = ax^3 + \tilde{b}x^2 + \tilde{c}x + \tilde{d} = 0$$ where $$\tilde{b}, \tilde{c}, \tilde{d}$$ are random such that $$\mathbb{E}[\tilde{b}] = b, \mathbb{E}[\tilde{c}] =c, \mathbb{E}[\tilde{d}] = d$$, and $$\tilde{b}, \tilde{c}, \tilde{d}$$ have known and finite variances. I wonder, if $$x$$ is a root to $$p_1$$, and $$\tilde{x}$$ is a root to $$p_2$$, then can we say anything on the relation between $$x$$ and $$\mathbb{E}[\tilde{x}]$$? Is $$\mathbb{E}[\tilde{x}] = x$$? If not, can we write $$\mathbb{E}[\tilde{x}]$$ in terms of the variance of coefficients?

Since there are multiple roots of a cubic polynomial, it may make above statements sound strange. But the basic idea is, if the polynomial with deterministic coefficients has a root at $$x$$, do we expect the corresponding polynomial with randomness in coefficients also have a root at $$x$$? Many thanks!

Is $$\mathbb{E}[\tilde{x}] = x$$?
No --- in the case of just $$x^3-d$$, the root is $$\sqrt[3]{d}$$, and $$E(\sqrt[3]{d})\ne\sqrt[3]{E(d)}$$ typically. For instance let $$d$$ be 1, 8, 27 with probability $$1/3$$ each, then $$E(\sqrt[3]{d})=2\ne\sqrt[3]{12}=\sqrt[3]{E(d)}.$$