The Critical Values of a Polynomial

Beardon, Alan F.; Carne, T. K.; Ng, Tuen Wai

doi:10.1007/s00365-002-0506-1

Cited by 23 publications

(136 citation statements)

References 7 publications

Supporting

Mentioning

135

Contrasting

Order By: Relevance

“…The injectivity of F p is an immediate consequence of Theorem 1. 1. The surjectivity claim in Theorem 1.2 is equivalent to the following Corollary 6.2 in the present paper, which follows from our work in Section 5.…”

Section: Level Curves and The Fingerprint Of A Shapesupporting

confidence: 55%

“…The steps for a counting proof of the full strength of Theorem 6.1 would be as follows. 1. Develop a notion of multiplicity for a member of P C a , (This seems very doable.)…”

Section: Possibilities For Extending the Results Of Lemma 51mentioning

confidence: 99%

“…We have two steps. 1. We choose some level k < n member ξ D P C of P C and assign it to D. This assignment must satisfy the following restrictions.…”

Section: The Possible Level Curve Configura-tions Of a Meromorphic Fumentioning

confidence: 99%

“…In a paper by Beardon, Carne, and Ng, [1] it was shown that for v ∈ V n−1 , if n = 2 then |H p,v | = 1, and if n ≥ 3 then H p,v contains exactly n n−3 elements according to multiplicity, where multiplicity arises through a use of Bezout's theorem. As an easy corollary to what was shown in that paper, one may prove that for v ∈ U n−1 , if n = 2 then |H p,v | = 1, and if n ≥ 3, then H p,v contains exactly n n−3 distinct members (that is, multiplicity plays no role).…”

Section: Partial Surjectivity Lemmamentioning

confidence: 99%

See 3 more Smart Citations

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Richards

2015

Comput. Methods Funct. Theory

View full text Add to dashboard Cite

Let f = B 1 /B 2 be a ratio of finite Blaschke products having no critical points having no critical points on ∂D. Then f has finitely many critical level curves (level curves containing critical points of f ) in the disk, and the non-critical level curves of f interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of f , one need only understand the configuration of the finitely many critical level curves of f . In this paper we show that in fact such a function f is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if f 1 and f 2 have the same configuration of critical level curves, then there is a conformal map φ such that f 1 ≡ f 2 •φ. We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss-Lucas theorem to rational functions) using level curves.Corollary: 6.2 For every finite Blaschke product B of degree n, there is some n degree polynomial p such that the set G := {z : |p(z)| < 1} is connected, and some conformal map φ : D → G such that B ≡ p • φ on D.

show abstract

Section: Level Curves and The Fingerprint Of A Shapesupporting

confidence: 55%

“…The steps for a counting proof of the full strength of Theorem 6.1 would be as follows. 1. Develop a notion of multiplicity for a member of P C a , (This seems very doable.)…”

Section: Possibilities For Extending the Results Of Lemma 51mentioning

confidence: 99%

“…We have two steps. 1. We choose some level k < n member ξ D P C of P C and assign it to D. This assignment must satisfy the following restrictions.…”

Section: The Possible Level Curve Configura-tions Of a Meromorphic Fumentioning

confidence: 99%

Section: Partial Surjectivity Lemmamentioning

confidence: 99%

See 2 more Smart Citations

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Richards

2015

Comput. Methods Funct. Theory

View full text Add to dashboard Cite

show abstract

“…. , z s−1 ) ∈ C s−1 |z i = 0 and z i = z j if i = j}/S s−1 via the map θ that sends each such polynomial to its set of critical values [4]. Let H be a homotopy of loops in S 1 ×V such that H starts at a loop x ∈ S 1 ×V which is in the image of (id, θ ) : S 1 ×W → S 1 ×V , where id is the identity map.…”

mentioning

confidence: 99%

Constructing a polynomial whose nodal set is any prescribed knot or link

Bode

Dennis

2019

J. Knot Theory Ramifications

View full text Add to dashboard Cite

We describe an algorithm that for every given braid B explicitly constructs a function f : C 2 → C such that f is a polynomial in u, v and v and the zero level set of f on the unit three-sphere is the closure of B. The nature of this construction allows us to prove certain properties of the constructed polynomials. In particular, we provide bounds on the degree of f in terms of braid data. I. INTRODUCTIONThere are various settings in which links can arise in zero level sets of polynomials. Among these the most prominent one is the study of links of isolated singularities [27]. While only a very restricted class of links can be the link of an isolated singularity, for some types of polynomials every link is possible. For example for any link L there exists a polynomial f : R 3 → C in three real variables with complex coefficients such that f −1 (0) is L. Even for these types explicit constructions are known for only very few examples. Dealing with explicit functions can be advantageous for studying some aspects of knot theory more closely. The critical points of the argument arg( f ) : S 3 \L → S 1 for example relate to circle-valued Morse and Novikov theory, which then connects to Reidemeister torsion and Seiberg-Witten invariants [20]. Explicit polynomials are also necessary for applications in the construction of knotted field configurations in physics. The goal of this paper is to describe a construction of such polynomials that works for any link.Knots as vanishing sets of polynomials have been mostly studied in the context of isolated singularities of a polynomial C 2 → C. For all small ε > 0 the intersection of the zero level set with the three-sphere of radius ε around the singularity is ambient isotopic to the same link, the link of the singularity [27]. For the links which arise as links of isolated singularities of complex polynomials, certain iterated cables of torus links [10,12,13,21,23,36], one can explicitly write down the corresponding polynomial. However, this procedure only covers a very restricted class of links. For a more detailed account of this topic we point the reader to [18,27].Similar considerations apply to polynomials R 4 → C with (weakly) isolated singularities [1,27], but while some knots like the figure-eight knot have been explicitly constructed in this setting [24,29,31], a constructive method that allows to generate polynomials with a zero level set of any given link explicitly is still missing. Akbulut and King showed in [1] that all links arise as links of weakly isolated singularities of real polynomials. However, their proof does not offer an explicit construction of these functions either.Knots have also been studied as real algebraic curves in RP 3 in an area that is now known as real algebraic knot theory. The knots that arise in this way for polynomials of degree less than or equal to five or equal to six with genus at most one have been classified [6,26].It follows from results in real algebraic geometry [1,7] that for any link L there exists a polynomial in three real ...

show abstract

Polynomials Versus Finite Blaschke Products

Tsang

2013

Fields Institute Communications

View full text Add to dashboard Cite

show abstract

The Critical Values of a Polynomial

Cited by 23 publications

References 7 publications

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Constructing a polynomial whose nodal set is any prescribed knot or link

Polynomials Versus Finite Blaschke Products

Contact Info

Product

Resources

About