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!


1 Answer 1


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 $$E(\sqrt[3]{d})=2\ne\sqrt[3]{12}=\sqrt[3]{E(d)}.$$


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