scholarly journals Critical points, critical values, and a determinant identity for complex polynomials

2020 ◽  
Vol 148 (12) ◽  
pp. 5277-5289
Author(s):  
Michael Dougherty ◽  
Jon McCammond
Keyword(s):  
Critical Points ◽  
Critical Values ◽  
Complex Polynomials

Affine cubic functions. IV. Functions on C 3 , nonsingular at infinity

1981 ◽  
Vol 302 (1470) ◽  
pp. 415-455 ◽  
Keyword(s):  
Detailed Analysis ◽  
Critical Points ◽  
Final Section ◽  
Critical Values ◽  
Complete List ◽  
Level Surface ◽  
Cubic Surface ◽  
Cubic Curve ◽  
Analysis Of Cases ◽  
Plane At Infinity

The object of this paper is to classify cubic functions f on C 3 according to their singularities. A level surface of such a function extends to a cubic surface in projective 3-space. The intersections S^,Tœ of S and its Hessian quartic T with the plane at infinity are the same for all levels. We assume throughout that is a nonsingular cubic curve. In §3 we show how the equisingularity class of Tœ determines the number and multiplicities of critical points of f .In § 2 we investigate n T^, and show that the equisingularity class of the pair (S^Tœ) determines that of f. Next we study the case when some point of has polar quadric a plane-pair; complete enumerations are given in §5 for the case when contains a line, and in §6 for when it contains an Eckardt point of S. In the final section we give a detailed analysis of cases when f has just two critical values, and show how to obtain a complete list of types of functions f.

Baker Domains of Period Two for the Family $f_{\lambda,\mu}(z)=\lambda e^{z} + \frac{\mu}{z}$

2014 ◽  
Vol 24 (05) ◽  
pp. 1450059 ◽  
Author(s):  
Marco A. Montes de Oca Balderas ◽  
Guillermo J. F. Sienra Loera ◽  
Jefferson E. King Davalos
Keyword(s):  
Critical Points ◽  
Critical Values ◽  
Fatou Set ◽  
The Family

Our main theorem establishes that the Fatou set of the functions fλ,μ(z) = λez + μ/z contains a two-cycle of Baker domains {F∞, F0} if Re(λ) < 0 and |Im(λ)| < 1/2|Re(λ)|. We show that under [Formula: see text] every point in F∞ tends to infinity and in F0 tends to zero. Moreover, if |Im(λ)| < 1/2|Re(λ)| - 4, the set F∞ contains infinitely many critical points of fλ,μ and F0 contains infinitely many critical values; also, infinitely many critical values of [Formula: see text] are contained in both F∞ and F0. Finally, the images of the Baker domains are displayed for some parameters.

The Situation over ℤ: Results

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Critical Points ◽  
Monic Polynomial ◽  
Critical Values

This chapter treats an integer monic polynomial f(x) ɛ ℤ[x] of degree n ≤ 2 which, over ℂ, is “weakly supermorse,” meaning that it has n distinct roots in ℂ, its derivative f'(x) has n − 1 distinct roots (the critical points) α‎ᵢ ɛ ℂ, and the n − 1 values f(α‎ᵢ) (the critical values) are all distinct in ℂ.

Bifurcation Analysis and Pattern Selection of Solutions for the Modified Swift–Hohenberg Equation

2020 ◽  
Vol 30 (11) ◽  
pp. 2030031
Author(s):  
Yuncherl Choi ◽  
Taeyoung Ha ◽  
Jongmin Han
Keyword(s):  
Bifurcation Analysis ◽  
Critical Points ◽  
Trivial Solution ◽  
Numerical Results ◽  
Critical Values ◽  
Pattern Selection ◽  
Periodic Condition ◽  
Pattern Formations ◽  
Selection Of

In this paper, motivated by [Peletier & Rottschäfer, 2004; Peletier & Williams, 2007], we study the dynamical bifurcation of the modified Swift–Hohenberg equation endowed with an evenly periodic condition on the interval [Formula: see text]. As [Formula: see text] crosses over the critical points, the trivial solution bifurcates to an attractor and some new patterns of solutions emerge. We provide detailed descriptions of all possible final patterns of solutions on the overlapped intervals of [Formula: see text], which emerge after a gap collapses to a point. We also compute all critical values of [Formula: see text], [Formula: see text] and [Formula: see text] precisely, which are responsible for bifurcation and pattern formations. We finally provide numerical results that explain the main theorems.

scholarly journals On Critical Values of Polynomials with Real Critical Points

2009 ◽  
Vol 32 (2) ◽  
pp. 385-392 ◽  
Author(s):  
Aimo Hinkkanen ◽  
Ilgiz Kayumov
Keyword(s):  
Critical Points ◽  
Critical Values

scholarly journals Principal Submatrices, Restricted Invertibility, and a Quantitative Gauss–Lucas Theorem

2018 ◽  
Vol 2020 (15) ◽  
pp. 4809-4832
Author(s):  
Mohan Ravichandran
Keyword(s):  
Convex Hull ◽  
Critical Points ◽  
Stable Rank ◽  
Complex Polynomials ◽  
Alternate Proof ◽  
Quantitative Version ◽  
Principal Submatrices ◽  
The Way

Abstract We apply the techniques developed by Marcus, Spielman, and Srivastava, working with principal submatrices in place of rank-$1$ decompositions to give an alternate proof of their results on restricted invertibility. This approach recovers results of theirs’ concerning the existence of well-conditioned column submatrices all the way up to the so-called modified stable rank. All constructions are algorithmic. The main novelty of this approach is that it leads to a new quantitative version of the classical Gauss–Lucas theorem on the critical points of complex polynomials. We show that for any degree $n$ polynomial $p$ and any $c \geq 1/2$, the area of the convex hull of the roots of $p^{(\lfloor cn \rfloor )}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.

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