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

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.


2014 ◽  
Vol 24 (05) ◽  
pp. 1450059 ◽  
Author(s):  
Marco A. Montes de Oca Balderas ◽  
Guillermo J. F. Sienra Loera ◽  
Jefferson E. King Davalos

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.


Author(s):  
Nicholas M. Katz

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.


2020 ◽  
Vol 30 (11) ◽  
pp. 2030031
Author(s):  
Yuncherl Choi ◽  
Taeyoung Ha ◽  
Jongmin Han

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.


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

2018 ◽  
Vol 2020 (15) ◽  
pp. 4809-4832
Author(s):  
Mohan Ravichandran

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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