Gauss-Lucas Theorem Research Articles

When D-companion matrix meets incomplete polynomials

In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we obtain a majorization relationship between the zeros of convex combinations of incomplete polynomials and the original polynomial. Moreover, we prove that the set of zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. Location of zeros of convex combinations of incomplete polynomials is determined.

Distribution of Zeros and Critical Points of a Polynomial, and Sendov’s Conjecture

According to the Gauss–Lucas theorem, the critical points of a complex polynomial p(z):=sum_{j=0}^{n}a_{j}z^{j} where a_{j}inmathbb{C} always lie in the convex hull of its zeros. In this paper, we prove certain relations between the distribution of zeros of a polynomial and its critical points. Using these relations, we prove the well-known Sendov’s conjecture for certain special cases.

Stability of matrix polynomials in one and several variables

The paper presents methods for the eigenvalue localisation of regular matrix polynomials, in particular, the stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Szász inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several seconds- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable.

Gauss Lucas theorem and Bernstein-type inequalities for polynomials

Abstract According to Gauss-Lucas theorem, every convex set containing all the zeros of a polynomial also contains all its critical points. This result is of central importance in the geometry of critical points in the analytic theory of polynomials. In this paper, an extension of Gauss-Lucas theorem is obtained and as an application some generalizations of Bernstein-type polynomial inequalities are also established.

On geometry of p-adic polynomials

In this paper, an analogue of the Gauss–Lucas theorem for polynomials over the algebraic closure [Formula: see text] of the field of [Formula: see text]-adic numbers is considered.

The strong Gauss–Lucas theorem and analyticity of correlation functions via the Lee–Yang theorem

We provide a simple mechanism for going from Lee–Yang type theorems to analyticity of correlation functions by exploiting under-appreciated inequalities of Newman. We also describe a Lee–Yang approach that recovers the consequences of a low density cluster expansion for spin S models without any combinatorics.

A quantitative Gauss–Lucas theorem

A quantitative Gauss–Lucas theorem

On the location of zeros of generalized derivative

Let $P(z) =displaystyle prod_{v=1}^n (z-z_v),$ be a monic polynomial of degree $n$, then, $G_gamma[P(z)] = displaystyle sum_{k=1}^n gamma_k prod_{{v=1},{v neq k}}^n (z-z_v),$ where $gamma= (gamma_1,gamma_2,dots,gamma_n)$ is a n-tuple of positive real numbers with $sum_{k=1}^n gamma_k = n$, be its generalized derivative. The classical Gauss-Lucas Theorem on the location of critical points have been extended to the class of generalized derivativecite{g}. In this paper, we extend the Specht Theorem and the results proved by A.Aziz cite{1} on the location of critical points to the class of generalized derivative .

Refinements of the Gauss-Lucas theorem using rational lemniscates and polar convexity

The classical Gauss-Lucas theorem for complex polynomials p ( z ) ≔ ( z − z 1 ) ⋯ ( z − z n ) p(z) ≔(z-z_1) \cdots (z-z_n) states that the critical points of p ( z ) p(z) are in the convex hull of its zeros. We give two refinements of the Gauss-Lucas theorem, both connected by the notion of polar convexity. In the first, we describe lemniscatic regions that do not contain non-trivial critical points of any polynomial of the form ( z − z 1 ) ⋯ ( z − z m ) ( z − z m + 1 ∗ ) ⋯ ( z − z n ∗ ) (z-z_1) \cdots (z-z_m)(z-z^*_{m+1})\cdots (z-z^*_{n}) , when the parameters z m + 1 ∗ , … , z n ∗ z^*_{m+1},\ldots ,z^*_{n} vary freely in a specified set, containing the zeros z m + 1 , … , z n z_{m+1},\ldots ,z_{n} . In the second, we locate the non-trivial critical points of p ( z ) p(z) in the intersection of m + 1 m+1 polar-convex sets, where m m is the number of distinct zeros of p ( z ) p(z) . This refinement complements recent ones in Dimitrov [Proc. Amer. Math. Soc. 126 (1998), pp. 2065–2070] and in Curgus and Mascioni [Proc. Amer. Math. Soc. 132 (2004), pp. 2973–2981].

Some recent results on the geometry of complex polynomials: the Gauss–Lucas theorem, polynomial lemniscates, shape analysis, and conformal equivalence

In this article, we survey the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are (i) Generalizations of the Gauss–Lucas Theorem, (ii) Geometry of Polynomials Level Sets, and (iii) Shape Analysis and Conformal Equivalence.

Sector Analogue of the Gauss–Lucas Theorem

AbstractThe classical Gauss–Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows.We show that if the coefficients of a monic polynomial $p(z)$ are in the sector {tei𝜓 : 𝜓∈ [0, 𝜙], t⩾0}, for some $\unicode[STIX]{x1D719}\in [0,\unicode[STIX]{x1D70B})$, and the zeros are not in its interior, then the critical points of $p(z)$ are also not in the interior of that sector.In addition, we give a necessary condition for a polynomial to satisfy the premise of the main result.

The location of critical points of polynomials

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials.

A STABILITY VERSION OF THE GAUSS–LUCAS THEOREM AND APPLICATIONS

Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk, $$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$ then $p^{\prime }$ has $n-1$ roots inside the unit disk and $m$ roots at distance at least $(dn-m)/(n+m)>1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for $n$ sufficiently large, each of the $m$ roots has a critical point at distance ${\sim}n^{-1}$.

Boundary Gauss–Lucas type theorems on the disk

The classical Gauss—Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We consider similar phenomena for generic inner functions, as well as for certain “locally inner” self-maps of the disk. More precisely, we look at a unit-norm function f ∈ H∞ that has an angular derivative on a set of positive measure (on the boundary) and we assume that its inner factor, I, is nontrivial. Under certain conditions to be discussed, it follows that f′ must also have a nontrivial inner factor, say J, and we study the relationship between the boundary singularities of I and J. Examples are furnished to show that our sufficient conditions cannot be substantially relaxed.

Geometry of a Family of Complex Polynomials

Let Pa be the family of complex-valued polynomials of the form p(z)=(z-a)(z-r)(z-s) with a in [0,1] and r and s on the unit circle. The Gauss-Lucas Theorem implies that the critical points of a polynomial in Pa lie in the unit disk. This paper characterizes the location and structure of these critical points. We show that the unit disk contains an open circular disk in which critical points of polynomials in Pa do not occur. Furthermore, almost every c inside the unit disk and outside of the desert region is the critical point of a unique polynomial in Pa.

Leaky roots and stable Gauss-Lucas theorems

ABSTRACTLet be a polynomial. The Gauss-Lucas theorem states that its critical points, , are contained in the convex hull of its roots. In a recent quantitative version, Totik shows that if almost all roots are contained in a bounded convex domain , then almost all roots of the derivative are in an neighborhood (in a precise sense). We prove a quantitative version: if a polynomial p has n roots in K and roots outside of K, then has at least n−1 roots in . This establishes, up to a logarithm, a conjecture of the first author: we also discuss an open problem whose solution would imply the full conjecture.

Speiser’s Theorem on the Road

In this note we discuss the Gauss-Lucas theorem (for the zeros of the derivative of a polynomial) and Speiser’s equivalent for the Riemann hypothesis (about the location of zeros of the Riemann zeta-function). We indicate similarities between these results and present there analogues in the context of elliptic curves, regular graphs, and finite Euler products.

Convolutions of Harmonic Right Half-Plane Mappings with Harmonic Strip Mappings

We prove that convolutions of harmonic right half-plane mappings with harmonic vertical strip mappings are univalent and convex in the horizontal direction. The proofs of these results make use the Gauss–Lucas Theorem. Our results show that two recent conjectures, the one by Kumar, Gupta, Singh and Dorff, and the one of Liu, Jiang and Li, are true. Moreover, examples of univalent harmonic mappings related to the above-mentioned results are presented, suggesting that the bounds given by our results may be sharp.

How ( Δ<i>t</i> )<sup>5</sup> + <i>A</i><sub>1</sub> ⋅ ( Δ<i>t</i> )<sup>2</sup> + <i>A</i><sub>2</sub> = 0 Is Generally, in the Galois Sense Solvable for a Kerr-Newman Black Hole Affect Questions on the Opening and Closing of a Wormhole Throat and the Simplification of the Problem, Dramatically Speaking, If <i>d</i> = 1 (Kaluza Klein Theory) and Explaining the Lack of Overlap with the

First off, the term Δt is for the smallest unit of time step. Now, due to reasons we will discuss we state that, contrary to the wishes of a reviewer, the author asserts that a full Galois theory analysis of a quintic is mandatory for reasons which reflect about how the physics answers are all radically different for abbreviated lower math tech answers to this problem. i.e. if one turns the quantic to a quadratic, one gets answers materially different from when one applies the Gauss-Lucas theorem. So, despite the distaste of some in the physics community, this article pitches Galois theory for a restricted quintic. We begin our analysis of if a quintic equation for a shift in time, as for a Kerr Newman black hole affects possible temperature values, which may lead to opening or closing of a worm hole throat. Following Juan Maldacena, et al., we evaluate the total energy of a worm hole, with the proviso that the energy of the worm hole, in four dimensions for a closed throat has energy of the worm hole, as proportional to negative value of (temperature times a fermionic number, q) which is if we view a worm hole as a connection between two black holes, a way to show if there is a connection between quantization of gravity, and if the worm hole throat is closed. Or open. For the quantic polynomial, we relate Δt to a ( Δt )5 + A1 ⋅ ( Δt )2 + A2 = 0 Quintic polynomial which has several combinations which Galois theoretical sense is generally solvable. We find that A2 has a number, n of presumed produced gravitons, in the time interval Δt and that both A1 and A2 have an Ergosphere area, due to the induced Kerr-Newman black hole. If Gravitons and Gravitinos have the relationship the author purports in an article the author wrote years ago, as cited in this publication, then we have a way to discuss if quantization of gravity as affecting temperature T, in the worm hole tells us if a worm hole is open or closed. And a choice of the solvable constraints affects temperature, T, which in turn affects the sign of a worm hole throat is far harder to solve. We explain the genesis of black hole physics negative temperature which is necessary for a positive black hole entropy, and then state our results have something very equivalent in terms of worm ding ( Δt )5 + A1 ⋅ ( Δt )2 + A2 = 0 we will be having X = Δt assumed to be negligible, We then look at a quadratic version in the solution of X = Δt so we are looking at four regimes for solving a quintic, with the infinitesimal value of Δt effectively reduced our quintic to a quadratic equation. Note that in the small Δt limit for d = 1, 3, 5, 7, we cleanly avoid any imaginary time no matter what the sign of Ttemp is. In the case where we have X = Δt assumed to be negligible, the connection in our text about coupling constants, if d = 3, may in itself for infinitesimal Δt lend toward supporting d = 3. This is different from the more general case for general Galois solvability of ( Δt )5 + A1 ⋅ ( Δt )2 + A2 = 0. d ≠ 1 means we need to consider Galois theory. If d = 2, 4, 6, need Ttemp A1 to be greater than zero. If d ≠ 1 and is instead d = 3, 5, 7, there is an absence of general solutions in the Galois solution sense. This because if. d ≠ 1 A1 X = Δt assumed to be infinitesimal in ( Δt )5 + A1 ⋅ ( Δt )2 + A2 = 0.

Fixed points of mapping of N-point gravitational lenses

In this paper, we study fixed points of N-point gravitational lenses. We use complex form of lens mapping to study fixed points. Complex form has an advantage over coordinate one because we can describe N-point gravitational lens by system of two equation in coordinate form and we can describe it by one equation in complex form. We can easily transform the equation, which describe N-point gravitational lens, into polynomial equation that is convenient to use for our research. In our work, we present lens mapping as a linear combination of two mapping: complex analytical and identity mapping. Analytical mapping is specified by analytical function (deflection function). We studied necessary and sufficient conditions for the existence of deflection function and proved some theorems. Deflection function is analytical, rational, its zeroes are fixed points of lens mapping and their number is from 1 to N-1, poles of deflection function are coordinates of point masses, all poles are simple, the residues at the poles are equal to the value of point masses.\\ We used Gauss-Lucas theorem and proved that all fixed points of lens mapping are in the convex polygon. Vertices of the polygon consist of point masses. We proved theorem that can be used to find all fixed point of lens mapping. On the basis of the above, we conclude that one-point gravitational lens has no fixed points, 2-point lens has only 1 fixed point, 3-point lens has 1 or 2 fixed points. Also we present expressions to calculate fixed points in 2-point and 3-point gravitational lenses. We present some examples of parametrization of point masses and distribution of fixed points for this parametrization.