Abstract
We consider random polynomials whose coefficients are independent and identically distributed on the integers. We prove that if the coefficient distribution has bounded support and its probability to take any particular value is at most $\tfrac12$, then the probability of the polynomial to have a double root is dominated by the probability that either $0$, $1$, or $-1$ is a double root up to an error of $o(n^{-2})$. We also show that if the support of coefficient distribution excludes $0$ then the double root probability is $O(n^{-2})$. Our result generalizes a similar result of Peled, Sen and Zeitouni for Littlewood polynomials.
Highlights
Let n ∈ N and let0≤j be a sequence of independent and identically distributed random variables taking values in Z
We extend the result for more general integer-valued random variables
(d) Even though some parts of our proof closely follow the lines of arguments from the paper of Peled, Sen and Zeitouni [13], extending the result to general integer-valued coefficients, poses a few significant challenges
Summary
Let n ∈ N and let (ξj )0≤j be a sequence of independent and identically distributed random variables taking values in Z. We extend the result for more general integer-valued random variables. (d) Even though some parts of our proof closely follow the lines of arguments from the paper of Peled, Sen and Zeitouni [13], extending the result to general integer-valued coefficients, poses a few significant challenges. One consequence of this difficulty is that the result here requires the maximal atom of the coefficient distribution to be at. There are algebraic numbers which are not a root of unity and all of its conjugates lie on the unit circle This requires new methods for handling such roots which are given in sections 5 and 6. A key instrument in the proof of the theorem, which may be of independent interest, is the following anti-concentration bound.
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