Polynomials In Complex Variables Research Articles

All links are semiholomorphic

Semiholomorphic polynomials are functions f:{mathbb {C}}^2rightarrow {mathbb {C}} that can be written as polynomials in complex variables u, v and the complex conjugate overline{v}. The origin is a weakly isolated singularity of a polynomial map if it is locally the only critical point on the variety. In this case the intersection of the variety and a sufficiently small 3-sphere produces a link whose link type is a topological invariant of the singularity. We prove the semiholomorphic analogue of Akbulut’s and King’s All knots are algebraic, that is, every link type in the 3-sphere arises as the link of a weakly isolated singularity of a semiholomorphic polynomial. Our proof is constructive, which allows us to obtain an upper bound on the polynomial degree of the constructed functions.

Construction on the Degenerate Poly-Frobenius-Euler Polynomials of Complex Variable

In this paper, we introduce degenerate poly-Frobenius-Euler polynomials and derive some identities of these polynomials. We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.

Some Identities of the Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable

In this paper, we introduce degenerate multi-poly-Bernoulli polynomials and derive some identities of these polynomials. We give some relationship between degenerate multi-poly-Bernoulli polynomials degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate multi-poly-Bernoulli polynomials of complex variables, and then, we derive several properties and relations.

Unification of the Nature's Complexities via a Matrix Permanent-Critical Phenomena, Fractals, Quantum Computing, ♯P-Complexity.

We reveal the analytic relations between a matrix permanent and major nature’s complexities manifested in critical phenomena, fractal structures and chaos, quantum information processes in many-body physics, number-theoretic complexity in mathematics, and ♯P-complete problems in the theory of computational complexity. They follow from a reduction of the Ising model of critical phenomena to the permanent and four integral representations of the permanent based on (i) the fractal Weierstrass-like functions, (ii) polynomials of complex variables, (iii) Laplace integral, and (iv) MacMahon master theorem.

Nimble evolution for pretzel Khovanov polynomials

We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables q and T, for pretzel knots of genus g in some regions in the space of winding parameters n_0, dots , n_g. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at Tne -1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots the two eigenvalues 1 and lambda = q^2 T, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and -q^2. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” lambda , namely, they are equal to lambda ^2, dots , lambda ^g. From point of view of frequencies, i.e. logarithms of eigenvalues, this is frequency doubling (more precisely, frequency multiplication) – a phenomenon typical for non-linear dynamics. Hence, our observation can signal a hidden non-linearity of superpolynomial evolution. To give this newly observed evolution a short name, note that when lambda is pure phase the contributions of lambda ^2, dots , lambda ^g oscillate “faster” than the one of lambda . Hence, we call this type of evolution “nimble”.

On the enhancement to the Milnor number of a class of mixed polynomials

The enhancement to the Milnor number is an invariant of the homotopy classes of fibered links in the sphere $S^{2n-1}$ and belongs to $\mathbb{Z}/r\mathbb{Z}$, where $r=0$ if $n=2$ and $r=2$ if $n=2$. Mixed polynomials are polynomials in complex variables $z_1,\dots,z_n$ and their conjugates $\bar{z}_1,\dots,\bar{z}_n$. M. Oka showed that mixed polynomials have Milnor fibrations under the strongly non-degeneracy condition. In this present paper, we study fibered links which are defined by a certain class of mixed polynomials which admit Milnor fibrations and show that any element of $\mathbb{Z}/r\mathbb{Z}$ is realized by the enhancement to the Milnor number of such a fibered link.

Algorithms for Computing Generalized U(N) Racah Coefficients

Algorithms are developed for computing generalized Racah coefficients for the U(N) groups. The irreducible representations (irreps) of the U(N) groups, as well as their tensor products, are realized as polynomials in complex variables. When tensor product irrep labels as well as a given irrep label are specified, maps are constructed from the irrep space to the tensor product space. The number of linearly independent maps gives the multiplicity. The main theorem of this paper shows that the eigenvalues of generalized Casimir operators are always sufficient to break the multiplicity. Using this theorem algorithms are given for computing the overlap between different sets of eigenvalues of commuting generalized Casimir operators, which are the generalized Racah coefficients. It is also shown that these coefficients are basis independent.

The molecular vibration-rotation Hamiltonian in the Bargmann-Hilbert space

The molecular vibration-rotation Hamiltonian of polyatomic molecules is developed in the Bargmann-Hilbert space of entire functions. The vibration-rotation eigenvectors of polyatomic molecules are described as polynomials in complex variables where the number of complex variables is equal to the number of degrees of freedom of the molecule. Simple differential operator representations are found for all rotational and vibrational operators. A special procedure is developed, in the new representation, for describing the direction cosine operators of a rigid symmetric rotor. The new basis of molecular states is shown to be very convenient for calculating energy or intensity perturbations of molecules with axial symmetry. The theoretical methods include a treatment of doubly degenerate modes of vibration.

A unification of boson expansion theories: (I). Functional representations of fermion states

The method for obtaining boson expansion by representing the fermion states as holomorphic functions of many complex variables is presented. Such functional representation is explicitly constructed for each space which is the carrier space of an irreducible representation of a semisimple compact Lie group. This is achieved by proving the unity resolution in terms of holomorphically parametrized Perelomov's generalized coherent states. The functional images of fermion states are polynomials of complex variables, while those of fermion operators are differential operators of finite order with polynomial coefficients.

ON THE APPROXIMATION OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES ON FAT COMPACT SUBSETS OF $ \mathbf{C}^n$ BY POLYNOMIALS

For a compact set , we denote by the algebra of all functions on which can be approximated uniformly (on ) by polynomials in complex variables, and by the algebra of all continuous functions on which are analytic at the interior points of . We shall say that is fat if it is the closure of an open set.In this paper, we consider the problem of approximating functions of several complex variables on fat compact sets with connected interior by polynomials. We prove the following theorems.THEOREM 1. There exists a fat polynomially convex (holomorphically) contractible compact subset of whose interior is homeomorphic to the four-dimensional open ball and such that .THEOREM 2. There exists a fat polynomially convex contractible compact subset of whose interior is homeomorphic to the six-dimensional open ball and such that , although the minimal boundaries of the algebras and coincide.Bibliography: 15 titles.