No ensemble-free extreme-root asymptotic and a Kac small-ball transfer theorem

Definitions and mathematical statement

For a random polynomial $P_n(z)=\sum_{j=0}^n a_{n,j}z^j$ of degree $n\geq1$ almost surely, define its largest-root modulus by \[ R_n=\max\{|z|:P_n(z)=0\}. \] For real coefficients this definition still takes all complex roots, not just real roots. Wenbo Li's question asks for asymptotics of $R_n$ as $n\to\infty$, for random polynomials in one real or complex variable. The verified source gives no coefficient law, weights, dependence assumptions, or mode of asymptotics. A precise theorem must specify the ensemble and whether it seeks a growth scale, convergence in probability or distribution, tails, or moments. A real-roots-only interpretation would be a different question and would need a convention when no real root exists.

The attached aid illustrates why there is no single ensemble-free scale. For independent standard real or circular complex Gaussian coefficients $\xi_j$, the Kac ensemble is $P_n(z)=\sum_{j=0}^n\xi_jz^j$, and its reported largest-root moduli are tight. The usual Gaussian Weyl ensemble instead has $W_n(z)=\sum_{j=0}^n\xi_jz^j/\sqrt{j!}$; the aid reports the lower bound \[ \Pr\left\{R_n(W_n)\geq(e^{-1/2}-\varepsilon)\sqrt n\right\}\longrightarrow1 \quad\text{for every }\varepsilon>0. \] This is a lower bound, not a complete Weyl extreme-root asymptotic.

For a more specific Kac claim, let the real or complex $\xi_j$ be independent and identically distributed, and suppose that for some $\alpha>0$, $u_0,c,C>0$, \[ c u^\alpha\leq\Pr\{|\xi_0|\leq u\}\leq C u^\alpha \quad(0<u\leq u_0),\qquad \mathbb E|\xi_0|^\alpha<\infty. \] In particular the coefficient law has no atom at zero. The aid claims constants $x_0,c_1,C_1>0$, depending on the law but not the degree, such that \[ c_1x^{-\alpha}\leq\Pr\{R_n>x\}\leq C_1x^{-\alpha} \quad(n\geq1,\ x\geq x_0). \] It reports an associated limiting random-power-series law $R_\infty$ with the same tail order, and convergence of every subcritical positive moment: \[ \sup_{n\geq1}\mathbb E R_n^s<\infty, \qquad \mathbb E R_n^s\longrightarrow\mathbb E R_\infty^s<\infty \quad(0<s<\alpha). \] For $s\geq\alpha$, the claimed lower tails force divergence of the corresponding moments at every finite degree and for the limiting law. The exact construction of the limiting power-series variable is not reproduced in the record. These are supplied unreviewed, ensemble-specific results; they do not assign an asymptotic to unspecified random polynomials.

Short English statement

Find the size and distribution of the largest-modulus zero as degree grows, after specifying a random-polynomial ensemble. The source specifies none. The supplied aid contrasts tight Gaussian Kac extremes with growing Weyl extremes and gives a conditional Kac tail and moment theorem based on coefficient small-ball behavior.

Sources

• Wenbo Li, Question 1, AIM workshop Random analytic functions: https://aimath.org/WWN/randomzeros/arcc1.pdf. The supplied verification says the PDF and following questions supply no ensemble definition.
• Local registry data/unsolvedmath/problems.json, permanent ID $20000626$, exact source code catalogue entry; canonical aim-analysis-notes.json, notes[157].
• The attached aid supplies the Kac/Weyl comparison and small-ball theorem without full external bibliography. Attempt 1 is explicitly machine-generated, not peer reviewed, and requires independent verification before citation.