The interior-maximum qualification in equation (33)

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Source. Bernstein 1931, equations (32)--(33), printed p. 1040 / PDF p. 16, with the setup on printed p. 1039 / PDF p. 15, in the complete source. Equation (33) displays

$$F(\xi)>\frac12\log n-O(\log\log\log n)$$

when the maximum of the nodal polynomial's modulus on a fixed interval is attained at an interior point. This page preserves that source claim and separates it from the version completely derived below.

A version with explicit uniform hypotheses. Fix $I=[\alpha,\beta]\subseteq[-1,1]$ of length $L>0$ and $0<\eta\le L/2$. For each degree $d$, suppose there is a point

$$\xi\in[\alpha+\eta,\beta-\eta], \qquad |A(\xi)|=\max_I|A|,$$

where $A$ is the nodal polynomial for $d+1$ distinct nodes in $[-1,1]$. Put $M=\max_I F$ and $m=\lfloor d/2\rfloor$. For all sufficiently large $d$, uniformly in the nodes,

$$M>\frac12\log\frac{\eta m}{2\log(2\log d)} =\frac12\log d-O_{I,\eta}(\log\log\log d). \tag{H}$$

In the case $M\le\log d$, the finite right side is also a lower bound for $F(\xi)$ itself.

Proof. If $M>\log d$, (H) is immediate for $d\ge16$, because $\eta\le1$, $m\le d/2$, and $\log(2\log d)>1$ make its right side less than $\log d$. Otherwise the local gap companion applies with

$$D=\frac{2\log(2M)}m \le D_0=\frac{2\log(2\log d)}m.$$

Take $d$ large enough that $D_0<\eta/2$. There is a first node $u<\alpha+D$ in $I$ and a last node $v>\beta-D$ in $I$. Hence $\xi-u>\eta/2$ and $v-\xi>\eta/2$. The consecutive nodes surrounding $\xi$ belong to this block and have gap $\delta<D$. Equation (T2) on the telescoping page therefore gives

$$F(\xi)> \frac14\log\frac{4(\xi-u)(v-\xi)}{\delta^2} >\frac12\log\frac{\eta}{D} \ge\frac12\log\frac{\eta}{D_0}.$$

This is (H). Its asymptotic form follows as on the all-cases local-bound page. All inequalities remain valid when $I$ touches $\pm1$.

Unresolved source implication. Equation (32) retains the factor $(\xi-a_p)(a_q-\xi)$. The assertion that $\xi$ is an interior point, for each degree separately, does not by itself give a degree-independent positive lower bound for these two distances. For example, the logical condition $\alpha<\xi_d<\beta$ permits $\xi_d-\alpha\to0$; this observation is not a counterexample involving nodal polynomials.

The source's displayed passage from (32) to (33) does not explicitly provide the additional uniform-distance estimate. This compilation does not prove or refute (33) under its bare interiority wording. The exact remaining obligation is to justify the necessary product control from the nodal-maximum hypotheses, or to state an appropriate additional hypothesis. The theorem (H) above uses the explicit sufficient hypothesis of uniform separation. It is not presented as an author-issued correction.

Proof scope. The uniformly separated version has a complete rewritten proof, reviewed on 6 September 2026 as component C7 of the local-chain review, which passed it conditionally on the (T1) correction now present in equation (32). The bare-interiority source implication and the claim at a selected nodal maximum in the large-$M$ case remain unresolved. Neither is needed for the completed $1/4$ local-maximum argument. No sharp $2/\pi$ or formal credit follows.

Bears on. Problem 1153, qualified historical bound.