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Konyagin and Schlag [KoSc99] prove that for a random polynomial $f$ of degree $n$ with independent uniform $\pm1$ coefficients and every $\varepsilon>0$,

$$\limsup_{n\to\infty}\mathbb{P}\bigl(m(f)\le\varepsilon n^{-1/2}\bigr)\le C\varepsilon$$

for an absolute constant $C$, where $m(f)=\min_{|z|=1}|f(z)|$. The minimum modulus is therefore not typically smaller than a constant times $n^{-1/2}$, which with Konyagin's upper bound $n^{-1/2+o(1)}$ (Konyagin 1994) gives $\varepsilon n^{-1/2}\le m(f)\le n^{-1/2+o(1)}$ for typical $f$ and shows that the exponent $-1/2$ in the second question of Problem 525 is optimal, as the paper's abstract states: the power $n^{-1/2}$ cannot be improved. The order of magnitude itself, an upper bound $O(n^{-1/2})$ for typical $f$, follows only from Cook and Nguyen's limit law. The paper is not held in the library; the statement is recorded as the publisher's abstract, the site's commentary and the introduction of Cook and Nguyen's paper (card) state it. The paper was received by the journal on 1997-02-05 and in revised form on 1997-09-24 and was published electronically on 1999-08-27, as the publisher's record states; the page is dated by the publication date.

Covers. The lower half of the second question: for every $\varepsilon>0$ the limiting proportion of degree-$n$ sign polynomials with $m(f)\le\varepsilon n^{-1/2}$ is at most $C\varepsilon$, so, with Konyagin's upper bound, the exponent $-1/2$ is optimal for the typical minimum modulus. The exact order and the limit law are Cook and Nguyen 2021. The claim value is proved: a bound proved.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: the paper appeared in Transactions of the American Mathematical Society 351 (1999), no. 12, 4963–4980. Reviewed: the site's curator, Thomas F. Bloom, records the bound as showing Konyagin's upper bound essentially best possible in the problem's commentary, and Cook and Nguyen's refereed paper of 2021 records the bound in its introduction. No formal proof of this bound on its own is held or audited in this repository, so no formalized evidence is listed.