Constrained-zero Widom-factor conjecture

Determine whether the modified Widom factors \(\mathcal W_n^*(\gamma)\), defined by minimizing the supremum norm of monic degree-\(n\) polynomials whose zeros all lie on an analytic Jordan arc \(\gamma\), satisfy \(\lim_{n\to\infty}\mathcal W_n^*(\gamma)=2\).

Background

The ordinary Widom factor studied in the paper allows arbitrary zeros and has a geometry-dependent limit for analytic Jordan arcs. The authors propose a constrained variant in which every zero is required to lie on the arc.

They ask whether this zero constraint restores Widom’s original predicted limiting value 2. The result is known for intervals and circular arcs, and the authors believe it should hold for general sufficiently smooth arcs, but they explicitly state that the lower bound requires new ideas.

References

Another intriguing question is whether Widom's conjecture holds in the original form eq:Widom-conj if we constrain the zeros to lie on the arc \gamma. More precisely, we define a modified Widom factor

\mathcal{W}n*(\gamma) = Cap(\gamma){-n} \inf{z_1,\ldots,z_n\in\gamma} \Big\lVert \prod_{j=1}n (z-z_j)\Big\rVert_{\gamma},

which corresponds to minimizing the norm of monic polynomials with all their zeros constrained to the arc \gamma. Does it hold that \lim_{n\to\infty}\mathcal{W}_n*(\gamma)=2?

— Chebyshev polynomials on a Jordan arc  (2608.13445 - Buchecker et al., 13 Aug 2026) in Section 2, Discussion