Narrow Results

In a previous article with A. Aftalion and X. Blanc, it was shown that the hypercontractivity property of the dilation semigroup in spaces of entire functions was a key ingredient in the study of the Lowest Landau Level model for fast rotating Bose–Einstein condensates. That former work was concerned with the stationary constrained variational problem. This article is about the nonlinear Hamiltonian dynamics and the spectral stability of the constrained minima with motivations arising from the description of Tkatchenko modes of Bose–Einstein condensates. Again the hypercontractivity property provides a very strong control of the nonlinear term in the dynamical analysis.

A polynomial P is called a strong uniqueness polynomial for the family of non-constant entire functions on the complex plane if one cannot find two distinct non-constant entire functions f and g and a non-zero constant c such that P(f)=cP(g). We give necessary and sufficient geometric conditions on the divisor of zeros of P that P be a strong uniqueness polynomial for the family of non-constant entire functions on the complex plane.

In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite nonzero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problems 1.19 and 1.20 in a book of Hayman and Lingham [Research Problems in Function Theory, preprint (2018), https://arxiv.org/pdf/1809.07200.pdf]. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials $Q(f)$ to have some terms of any degree of $f$ and also the hypothesis $n>k$ in [Theorem 2 of W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana11(2) (1995) 355–373] is replaced by $n\ge 2$ in our result.

We introduce a new class of entire functions ${ℰ}$ which consists of all $F_0\in {𝒪}(ℂ)$ for which there exists a sequence $(F_n)\in {𝒪}(ℂ)$ and a sequence $({\lambda }_n)\in ℂ$ satisfying $F_n(z)={\lambda }_{n+1}{e}^{F_{n+1}(z)}$ for all $n\ge 0$. This new class is closed under the composition and it is dense in the space of all nonvanishing entire functions. We prove that every closed set $V\subset ℂ$ containing the origin and at least one more point is the set of singular values of some locally univalent function in ${ℰ}$, hence, this new class has nontrivial intersection with both the Speiser class and the Eremenko–Lyubich class of entire functions. As a consequence, we provide a new proof of an old result by Heins which states that every closed set $V\subset ℂ$ is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally, we show that the class ${ℰ}$ contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.

We consider the family of entire transcendental maps given by F_λ,m(z)=λz^mexp(z) where m≥2. All functions F_λ,m have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].