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Statement

Definitions (pp. 142--143). A monic polynomial (1) is a $K$-polynomial if $E$ (the set where $|f|<1$) is connected, and a $\bar K$-polynomial if its closure $\bar E$ is connected. The paper notes that $f$ is a $K$-polynomial exactly when $|f|<1$ at every zero of $f'$, and a $\bar K$-polynomial exactly when $|f|\le1$ there. $K_n$ and $\bar K_n$ are these classes in degree $n$, $K_n^*$ is the set of $f\in\bar K_n$ with $|f|=1$ at every zero of $f'$, and $\mathcal D(f)$ is the discriminant of $f$.

Theorem 10 (p. 143). "If $f\in K_n$ $(n>1)$, then $|\mathcal D(f)|<n^n$; if $f\in\bar K_n$, then $|\mathcal D(f)|\leq n^n$, and the equality holds if and only if $f\in K_n^*$."

The paper says (p. 143) that the theorem establishes a conjecture raised by E. Netanyahu and proved independently by W. H. Fuchs (oral communication).

Remark (p. 143). "A similar argument shows that if $\bar E$ has $n$ components, then $|\mathcal D(f)|>n^n$."

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; the definitions on pp. 142--143, Theorem 10, its proof and the Remark on p. 143. The copy read is identified on the source card.

Read depth. Claims checked: the definitions, the theorem, the identity behind it and the Remark were read clause by clause on the page images of pp. 142--143 on 2026-10-08, and the identity was followed. Nothing here is independently reviewed.

Proof pointer

Page 143. Write $f'(z)=n\prod_{\nu=1}^{n-1}(z-z'_\nu)$. Then

$$|\mathcal D(f)|=\prod_{\nu<\mu}|z_\nu-z_\mu|^2=\prod_{\mu=1}^n|f'(z_\mu)| =n^n\prod_{\nu=1}^{n-1}|f(z'_\nu)|,$$

and the characterizations of $K_n$, $\bar K_n$ and $K_n^*$ through the values of $|f|$ at the critical points give the strict bound, the weak bound and the equality case. The paper writes out only the identity.

Dependencies

The critical-point characterization of $K$- and $\bar K$-polynomials (p. 142), which the paper justifies in one sentence: if $|f|<A$ at every zero of $f'$, then no lemniscate $|f|=B$ with $B\ge A$ has a multiple point, and conversely.

Bears on

• #1045: the problem's product $\prod_{i\ne j}|z_i-z_j|$ is $|\mathcal D(f)|$ for the monic polynomial with those zeros. Theorem 10 bounds it by $n^n$ over the class $\bar K_n$, defined by the critical values of $f$; the problem's class is defined by a diameter bound on the zeros, and the paper gives no implication between the two. The problem itself is Problem 13, posed directly after this theorem.