Mergelyan's theorem with zero-free polynomials

Mergelyan's theorem says that a continuous function on a compact plane set without holes, holomorphic inside, can be approximated uniformly by polynomials. If the function has no zeros inside the set, can the polynomials be chosen without zeros on the whole set? This innocent-looking strengthening is open, and Andersson showed it is equivalent to a strengthened universality property of the Riemann zeta function.

Problem

Let $K \subset \mathbb{C}$ be compact with connected complement $\mathbb{C} \setminus K$, and let $f$ be continuous on $K$, holomorphic on the interior of $K$, and with no zeros in the interior of $K$ (zeros on the boundary are allowed).

Question (Gauthier). Is there a sequence of polynomials $p_n$, none of which has a zero on $K$, such that $p_n \to f$ uniformly on $K$?

Without the zero-free requirement this is exactly Mergelyan's theorem.

Background

The question was recorded by Paul Gauthier at the AIM workshop on stability and hyperbolicity, and stated as a conjecture by Johan Andersson in 2010. Its interest comes from Voronin's universality theorem, which says that vertical translates $\zeta(s + it)$ of the Riemann zeta function approximate every nonvanishing holomorphic function on suitable compact sets in the strip $1/2 < \operatorname{Re} s < 1$. Andersson showed that the zero-free Mergelyan conjecture is equivalent to a version of Voronin universality in which the function only needs to be nonvanishing in the interior of $K$. Gauthier and Knese point out a connection with the Riemann hypothesis.

What is known

• The answer is yes when $K$ has no interior points (Andersson; independently Gauthier), when $K$ is strictly starlike (Gauthier), and when the interior of $K$ is a Jordan domain (Andersson, 2013).
• Gauthier and Knese (2012) gave a positive answer for a "chain" of Jordan domains.
• Andersson (2013) proved the equivalence between the general case and the corresponding universality conjecture for the zeta function, which is recorded next to this problem in the source as a conjecture of Andersson and Farmer.

What would count as progress

• A proof for compact sets whose interior has infinitely many components, or for other natural classes such as sets whose interior is a finite union of Jordan domains that touch.
• A counterexample: a compact set $K$ and a function $f$ that cannot be approximated by zero-free polynomials, which would also disprove the strengthened universality statement for $\zeta$.
• For a fixed explicit $K$ and $f$, numerical constructions of zero-free approximating polynomials, whose zero-freeness on $K$ can be certified with interval arithmetic or the argument principle.

References

1. J. Andersson, Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions, J. Approx. Theory 167 (2013), 201–210; arXiv:1010.0850.
2. P. M. Gauthier and G. Knese, Zero-free polynomial approximation on a chain of Jordan domains, Ann. Sci. Math. Québec 36 (2012), 107–112.
3. J. Falcó and P. M. Gauthier, Approximation in measure: Dirichlet problem, universality and the Riemann hypothesis, Izv. Math. 85 (2021), 547–561; arXiv:2002.10129.

Source

Imported from AIM Problem Lists hyperbolicpoly 9.1, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.