Wiener-Itô decomposition of polynomial operators formed with boson quasi-free fields

Orthogonal polynomials, as a generalized notion of multiple Wiener integrals, are constructed on non-commuting operators of free boson fields in non-Fock states. The orthogonal polynomials form a continuum of notions whose special cases are Wick products in Fock states and Hermite polynomials of commuting operators of free fields generally in non-Fock states. Structures of orthogonal polynomials as operators or operator-valued distributions are given, and multiplication formulas and commutation relations are presented.     

A family of cyclic cubic polynomials whose roots are systems of fundamental units

We give a family of cyclic cubic polynomials whose roots are systems of fundamental units of the splitting fields. These polynomials are constructed by a linear fractional transformation from Shanks’ polynomials with rational coefficients.     

Hereditary atomicity in integral domains

If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.     

Exact values of the sums of multiplicative characters of polynomials over finite fields

We give explicit constructions of polynomials of special form over finite fields the sum of whose multiplicative characters is exactly known.     

DISTRIBUTIONS OF LIMIT CYCLES FOR POLYNOMIAL VECTOR FIELDS (P_2, Q_3)

The distributions of limit cycles of cubic vector fields (P2, Q3) are considered in this paper, where P2 and Q3 are polynomials of x and y of order two and three, respectively. It is possibly seven different distributions of limit cycles given in [1]. We now prove that in which three kinds of distributions are impossible and other four kinds all can be realized by concrete vector fields of (P2,Q3). Some related results are also given.     

Smooth Fano Polytopes Whose Ehrhart Polynomial Has a Root with Large Real Part

The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this paper, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.     

Combinatorial Aspects of the K-Theory of Grassmannians

In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are representatives for the classes corresponding to the structure sheaves of Schubert varieties in the K-theory of Grassmannians. These Grothendieck polynomials are nonhomogeneous symmetric polynomials whose lowest homogeneous component is a Schur polynomial. Our treatment, which is closely related to the theory of Schur functions, gives new information about these polynomials. Our main results are concerned with the transition matrices between Grothendieck polynomials indexed by Grassmannian permutations and Schur polynomials on the one hand and a Pieri formula for these Grothendieck polynomials on the other.     

On center sets of ODEs determined by moments of their coefficients

The classical H. Poincaré Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coefficients of the field. In this paper we show that the set of centers in the Center-Focus problem can be determined as the set of zeros of some continuous functions from the moments of coefficients of this equation.     

Singular cycles of vector fields on regular parts of the boundary of Morse-Smale systems

At the boundary of the class of Morse-Smale vector fields there are vector fields whose unique degenerate phenomena is a singular cycle. We first characterize and classify all singular cycles which contains only one degeneracy (thesimple singular cycles: ssc). Each of these cycles defines a condimension one submanifold of vector fields. For some ssc its codimension one submanifold is a regular part of the boundary of the Morse-Smale systems. We characterize those ssc that defines this type of submanifold. Our ambient space isn dimensional,n2.Supported by Fondecyt, Proyecto 1930863.     

Local decoding and testing of polynomials over grids

We study the local decodability and (tolerant) local testability of low‐degree n‐variate polynomials over arbitrary fields, evaluated over the domain {0,1}ⁿ. We show that for every field there is a tolerant local test whose query complexity depends only on the degree. In contrast we show that decodability is possible over fields of positive characteristic, but not over the reals.     

Further results on a class of permutation polynomials over finite fields

A class of permutation polynomials with given form over finite fields is investigated in this paper, which is a further study on a recent work of Zha and Hu. Based on some particular techniques over finite fields, two results obtained by Zha and Hu are improved and new permutation polynomials are also obtained.     

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

Orthogonal polynomials on the unit circle and their derivatives

We characterize the sequences of orthogonal polynomials on the unit circle whose derivatives are also orthogonal polynomials on the unit circle. Some relations for the sequences of derivatives of orthogonal polynomials are provided. Finally, we pose some problems about orthogonality-preserving maps and differential equations for orthogonal polynomials on the unit circle.     

Ideals and graphs,Gröbner bases and decision procedures in graphs

The well known correspondence between even cycles of an undirected graph and polynomials in a binomial ideal associated to a graph is extended to odd cycles and polynomials in another binomial ideal. Other binomial ideals associated to an undirected graph are also introduced. The results about them with topics on monomial ideals are used in order to show decision procedures for bipartite graphs, minimal vertex covers, cliques, edge covers and matchings with algebraic tools. All such procedures are implemented in Maple 9.5.     

On Touchard polynomials

J. Touchard in his work on the cycles of permutations generalized the Bell polynomials in order to study some problems of enumeration of the permutations when the cycles possess certain properties.In the present paper (considering Touchards's generalization) we introduce and study a class of related polynomials. An exponential generating function, recurrence relations and connections with other well-known polynomials are obtained. In special cases, relations with Stirling number of the first and second kind, as well as with other numbers recently studied are derived. Finally, a combinatorial interpretation is discussed.     

Persymmetric determinants 5. Families of overlapping coaxial equivalent determinants

In an earlier paper the author constructed two infinite matrices and showed that they contain families of distinct submatrices whose determinants represent identical polynomials.

The object of this paper is to extend the earlier results in two directions. The first set of identities is closely related to the original set and may be regarded as a supplement to the original paper. The second set relates determinants whose elements are based partly on Stirling numbers of the Second Kind to polynomials whose terms are based on Stirling numbers of the First Kind.     

An Algebra of Exponential Polynomials as the Cofinite Dual of the Hopf Algebra of Polynomials

We prove that the cofinite dual of the Hopf algebra of polynomials in several variables can be represented as a Hopf algebra ? of exponential polynomials that contains the polynomials as a Hopf subalgebra. We also present some algebras isomorphic to ? whose elements are rational functions or multi-sequences.     

Permutation polynomials of degree 6 or 7 over finite fields of characteristic 2

In Dickson (1896–1897) [2], the author listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.     

Generalizations of the spectral theorem for matrices. II. matrix polynomials over arbitrary fields

The spectral theorem for matrices is generalized to matrix polynomials over arbitrary fields. An associated theory of generalized interpolation with respect to a set of prime polynomials is developed, and Drazin inverses are used to identify the spectral components. These results are then used to state the spectral theorem for functions of real matrices.     

On the singular values of Weber modular functions

The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants, extending the formulas of Gross-Zagier for the level 1 case.