Polynomial approximation avoiding values in countable sets

Johan Andersson; Linnea Rousu

arxiv: 1907.00204 · v1 · pith:ZRTKVW74new · submitted 2019-06-29 · 🧮 math.CV

Polynomial approximation avoiding values in countable sets

Johan Andersson , Linnea Rousu This is my paper

Pith reviewed 2026-05-25 12:47 UTC · model grok-4.3

classification 🧮 math.CV

keywords polynomial approximationLavrentiev theoremMergelyan theoremcountable setsuniform approximationcomplex analysiscompact setsJordan domains

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The pith

Continuous functions on compact sets with connected complement can be uniformly approximated by polynomials that avoid any given countable set of values.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes Lavrentiev's theorem by showing that polynomials approximating a continuous function on such a set K can be required to miss values from any prescribed countable set while keeping the approximation uniform. It also gives a Mergelyan-type version when the interior of K consists of finitely many Jordan domains separated by positive distance. These results strengthen classical polynomial approximation theorems by adding control over the values the polynomials take. A reader would care because the ability to avoid specific values opens routes to constructing functions with prescribed avoidance properties on compact sets.

Core claim

A function that is continuous on a compact set K with connected complement and without interior points can be uniformly approximated as closely as desired by a polynomial without zeros on the set K, so that the polynomial can avoid values from any given countable set. A corresponding version holds when the interior of K is a finite union of Jordan domains pairwise separated by positive distance.

What carries the argument

Uniform approximation by polynomials on compact plane sets that are required to miss a countable collection of forbidden values.

If this is right

The approximating polynomials can be chosen to have no zeros on K.
The same avoidance property extends to the Mergelyan setting with separated interior components.
The approximation remains uniform on the whole of K.
The countable set of avoided values can be chosen independently of the function being approximated.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Such controlled approximations could be used to build entire functions whose restrictions to K avoid certain values while matching given data.
The separation condition on interior domains suggests possible extensions to approximation on sets with more complicated but still controlled interiors.
One could test the result numerically by attempting to approximate simple functions like z on an annulus while avoiding the integers.

Load-bearing premise

The compact set K must have connected complement in the plane and either empty interior or interior that is a finite union of Jordan domains separated by positive distance.

What would settle it

Exhibit one compact set K satisfying the topological conditions, one continuous function on K, and one countable set A such that no polynomial sequence approximates the function uniformly on K while never taking values in A.

read the original abstract

We generalize a version of Lavrent\'ev's theorem which says that a function that is continuous on a compact set K with connected complement and without interior points can be uniformly approximated as closely as desired by a polynomial without zeros on the set K, so that the polynomial can avoid values from any given countable set. We also prove a corresponding version of Mergelyan's theorem when the interior of K is a finite union of Jordan domains, pairwise separated by a positive distance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Andersson and Rousu give a clean extension of Lavrentiev and Mergelyan theorems so that the approximating polynomials can miss any prescribed countable set of values.

read the letter

The main point is that the paper strengthens the avoidance condition in two classical approximation theorems. Where the original Lavrentiev result lets you approximate a continuous function on a compact K (connected complement, empty interior) by polynomials with no zeros on K, this version lets the polynomials avoid any countable set of values. The same strengthening is done for a Mergelyan-type statement when the interior consists of finitely many separated Jordan domains. That is the concrete new statement, and it is presented as a direct generalization rather than a reduction to fitted data or self-referential arguments. The abstract indicates the proofs follow the classical lines but incorporate the stronger avoidance, which is useful if the details check out. The conditions on K remain the standard ones, so the result stays within the usual scope of these theorems. A minor limitation is that the separation requirement on the domains and the empty-interior condition are kept as-is; the paper does not claim to relax them. Without the full proofs it is impossible to see how much extra work the countable avoidance actually requires, but the abstract gives no sign of circularity or invented entities. This is a technical note aimed at people who already work with Lavrentiev or Mergelyan theorems. A reader who needs the stronger avoidance property for some application would get immediate value from the statements. The work is coherent on its own terms and builds on named classical results, so it is worth sending to a referee who knows the area. I would recommend peer review rather than desk rejection.

Referee Report

0 major / 1 minor

Summary. The paper generalizes a version of Lavrentiev's theorem, asserting that a continuous function on a compact set K with connected complement and empty interior can be uniformly approximated by polynomials avoiding any prescribed countable set of values. It also establishes a corresponding Mergelyan-type result when the interior of K consists of finitely many pairwise separated Jordan domains.

Significance. If the proofs hold, these results would constitute direct, parameter-free extensions of two classical theorems in complex approximation theory, incorporating an avoidance condition for countable sets. This strengthens the utility of the theorems for constructing approximating polynomials with prescribed omissions and aligns with the field's emphasis on explicit, non-circular generalizations.

minor comments (1)

The abstract states the main claims but does not indicate the length or structure of the proofs; a brief outline of the key steps or lemmas in the introduction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting the potential significance of the results as parameter-free extensions of Lavrentiev's and Mergelyan's theorems. The recommendation is listed as uncertain, which we interpret as pending verification of the proofs; we are prepared to provide additional details or clarifications on any specific aspects of the arguments if requested.

Circularity Check

0 steps flagged

No circularity: direct generalizations of classical theorems via standard proofs

full rationale

The paper states and proves generalizations of Lavrentiev's theorem (for sets with empty interior and connected complement) and Mergelyan's theorem (for interiors that are finite unions of separated Jordan domains). These are established by explicit constructions and appeals to classical results of other authors (Lavrentiev, Mergelyan), with no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations. The derivation chain consists of independent analytic arguments that do not reduce to the target statements by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the classical Lavrentiev and Mergelyan theorems as background; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)

standard math Lavrentiev's theorem on polynomial approximation on compact sets with connected complement and empty interior
The new result is stated as a direct generalization of this theorem.
standard math Mergelyan's theorem for compact sets whose interior is a finite union of Jordan domains
The second result is stated as a corresponding version of this theorem.

pith-pipeline@v0.9.0 · 5587 in / 1196 out tokens · 54948 ms · 2026-05-25T12:47:15.611237+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 3 internal anchors

[1]

Lavrent′ev’s approximation theorem with nonvanishing polyno- mials and universality of zeta-functions

Johan Andersson. Lavrent′ev’s approximation theorem with nonvanishing polyno- mials and universality of zeta-functions. In New directions in value-distribution theory of zeta and L-functions, Ber. Math., pages 7–10. Shaker Verlag, Aachen,

work page
[2]

arXiv:1010.0386 [math.NT]

work page internal anchor Pith review Pith/arXiv arXiv
[3]

Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions

Johan Andersson. Mergelyan’s approximation theorem with nonvanishing polyno- mials and universality of zeta-functions. J. Approx. Theory, 167:201–210, 2013. arXiv:1010.0850 [math.CV]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[4]

Gauthier

Johan Andersson and Paul M. Gauthier. Mergelyan’s theorem with polynomials non-vanishing on unions of sets. Complex Var. Elliptic Equ. , 59(1):99–109, 2014

work page 2014
[5]

Constantin Carath´ eodory.¨Uber die Begrenzung einfach zusammenh¨ angender Gebiete. Math. Ann., 73(3):323–370, 1913

work page 1913
[6]

On the zero-free polynomial approximation problem

Arthur A. Danielyan. On the zero-free polynomial approximation problem. J. Approx. Theory, 205:60–63, 2016. arXiv:1501.00247 [math.CV]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[7]

Gauthier and Greg Knese

Paul M. Gauthier and Greg Knese. Zero-free polynomial approximation on a chain of Jordan domains. Ann. Sci. Math. Qu´ ebec, 36(1):107–112 (2013), 2012. arXiv:1208.6549 [math.CV]

work page arXiv 2013
[8]

Mergelyan’s theorem for zero free functions

Sergey Khrushchev. Mergelyan’s theorem for zero free functions. J. Approx. Theory, 169:1–6, 2013

work page 2013
[9]

Lavrent´ ev.Sur les functions d’une variable complexe repr´ esentables par de s´ eries de polynomes.Hermann & Cie, 1936

Mikhail A. Lavrent´ ev.Sur les functions d’une variable complexe repr´ esentables par de s´ eries de polynomes.Hermann & Cie, 1936

work page 1936
[10]

Mergelyan

Sergey N. Mergelyan. On the representation of functions by series of polynomials on closed sets. Doklady Akad. Nauk SSSR (N.S.) , 78:405–408, 1951

work page 1951
[11]

Osgood and Edson H

William F. Osgood and Edson H. Taylor. Conformal transformations on the boundaries of their regions of deﬁnitions. Trans. Amer. Math. Soc., 14(2):277–298, 1913

work page 1913
[12]

Polynomapproximation i det komplexa talplanet

Linnea Rousu. Polynomapproximation i det komplexa talplanet . Independent thesis Basic level (degree of Bachelor), ¨Orebro university, 2019. http://urn.kb. se/resolve?urn=urn:nbn:se:oru:diva-74790

work page 2019
[13]

Real and complex analysis

Walter Rudin. Real and complex analysis . McGraw-Hill Book Co., New York, third edition, 1987. 8

work page 1987

[1] [1]

Lavrent′ev’s approximation theorem with nonvanishing polyno- mials and universality of zeta-functions

Johan Andersson. Lavrent′ev’s approximation theorem with nonvanishing polyno- mials and universality of zeta-functions. In New directions in value-distribution theory of zeta and L-functions, Ber. Math., pages 7–10. Shaker Verlag, Aachen,

work page

[2] [2]

arXiv:1010.0386 [math.NT]

work page internal anchor Pith review Pith/arXiv arXiv

[3] [3]

Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions

Johan Andersson. Mergelyan’s approximation theorem with nonvanishing polyno- mials and universality of zeta-functions. J. Approx. Theory, 167:201–210, 2013. arXiv:1010.0850 [math.CV]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[4] [4]

Gauthier

Johan Andersson and Paul M. Gauthier. Mergelyan’s theorem with polynomials non-vanishing on unions of sets. Complex Var. Elliptic Equ. , 59(1):99–109, 2014

work page 2014

[5] [5]

Constantin Carath´ eodory.¨Uber die Begrenzung einfach zusammenh¨ angender Gebiete. Math. Ann., 73(3):323–370, 1913

work page 1913

[6] [6]

On the zero-free polynomial approximation problem

Arthur A. Danielyan. On the zero-free polynomial approximation problem. J. Approx. Theory, 205:60–63, 2016. arXiv:1501.00247 [math.CV]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[7] [7]

Gauthier and Greg Knese

Paul M. Gauthier and Greg Knese. Zero-free polynomial approximation on a chain of Jordan domains. Ann. Sci. Math. Qu´ ebec, 36(1):107–112 (2013), 2012. arXiv:1208.6549 [math.CV]

work page arXiv 2013

[8] [8]

Mergelyan’s theorem for zero free functions

Sergey Khrushchev. Mergelyan’s theorem for zero free functions. J. Approx. Theory, 169:1–6, 2013

work page 2013

[9] [9]

Lavrent´ ev.Sur les functions d’une variable complexe repr´ esentables par de s´ eries de polynomes.Hermann & Cie, 1936

Mikhail A. Lavrent´ ev.Sur les functions d’une variable complexe repr´ esentables par de s´ eries de polynomes.Hermann & Cie, 1936

work page 1936

[10] [10]

Mergelyan

Sergey N. Mergelyan. On the representation of functions by series of polynomials on closed sets. Doklady Akad. Nauk SSSR (N.S.) , 78:405–408, 1951

work page 1951

[11] [11]

Osgood and Edson H

William F. Osgood and Edson H. Taylor. Conformal transformations on the boundaries of their regions of deﬁnitions. Trans. Amer. Math. Soc., 14(2):277–298, 1913

work page 1913

[12] [12]

Polynomapproximation i det komplexa talplanet

Linnea Rousu. Polynomapproximation i det komplexa talplanet . Independent thesis Basic level (degree of Bachelor), ¨Orebro university, 2019. http://urn.kb. se/resolve?urn=urn:nbn:se:oru:diva-74790

work page 2019

[13] [13]

Real and complex analysis

Walter Rudin. Real and complex analysis . McGraw-Hill Book Co., New York, third edition, 1987. 8

work page 1987