Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

On zeros of polynomials orthogonal over a convex domain

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain.     

On the zeros of polynomials

In this paper we extend a classical result due to Cauchy and its improvement due to Datt and Govil to a class of lacunary type polynomials.     

On the isomorphism problem for finite dimensional diagram algebras

somorphism problems for finite dimensional algebras can be computationally hard. When the algebras are monomial, it is shown, refining work of Shirayanagi, that there is a simple definitive combinatorial method. However, examples show that no such criterion is possible if the class of algebras is expanded to that of diagram algebras (in the sense of Fuller). The presentation of a diagram algebra is field independent but the existence of an isomorphism between two such is not. (Subject classes: 16G30, 16P10, 20M25).     

On extending real functions to finite dimensional algebras

In a finite dimensional real algebra, we define functional calculus for real functions. The analytic hypothesis is replaced by a differentiable hypothesis. High order differentibility is essentially necessary and sufficient to our discussions. Moreover, we give some examples. Partially supported by NSF of China The authors gratefully acknowledge the support of K. C. Wong Education Foundation, Hong Kong     

On coefficients of polynomials over finite fields

In this paper we study the relation between coefficients of a polynomial over finite field Fq and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m|q−1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m?q−1 and . As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q3/2) operations.     

On the zeros of Meixner polynomials

We investigate the zeros of a family of hypergeometric polynomials $M_n(x;\beta ,c)=(\beta )_n\,{}_2F_1(-n,-x;\beta ;1-\frac{1}{c})$ , $n\in \mathbb N ,$ known as Meixner polynomials, that are orthogonal on $(0,\infty )$ with respect to a discrete measure for $\beta >0$ and $0<c<1.$ When $\beta =-N$ , $N\in \mathbb N $ and $c=\frac{p}{p-1}$ , the polynomials $K_n(x;p,N)=(-N)_n\,{}_2F_1(-n,-x;-N;\frac{1}{p})$ , $n=0,1,\ldots , N$ , $0<p<1$ are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials $M_n(x;\beta ,c)$ , $c<0$ and $n<1-\beta $ , the quasi-orthogonal polynomials $M_n(x;\beta ,c)$ , $-k<\beta <-k+1$ , $k=1,\ldots ,n-1$ and $0<c<1$ or $c>1,$ as well as the polynomials $K_{n}(x;p,N)$ with non-Hermitian orthogonality for $0<p<1$ and $n=N+1,N+2,\ldots $ . We also show that the polynomials $M_n(x;\beta ,c)$ , $\beta \in \mathbb R $ are real-rooted when $c\rightarrow 0$ .     

On the zeros of exponential polynomials

Suppose r = (r₁, …, r_M), r_j ? 0, γ_kj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γ_k · r = ∑_jγ_kjr_j. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑_{j = 1}^Na_je^{?λγ_j · r}, where each a_j is a nonzero real number. More specifically, if $\text{Z}\text{?}\text{(a, r) =}\text{closure}\text{{}\text{Re}\text{λ: h(λ, a, r) = 0}}$, we study the dependence of $\text{Z}\text{?}\text{(a, r)}\text{on}\text{a, r}$. This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when $\text{0 ?}\text{Z}\text{?}\text{(a, r)}$ are given. Several examples illustrate the complicated nature of $\text{Z}\text{?}\text{(a, r)}$. The results have immediate implication to the theory of stability for difference equations x(t) ? ∑_{k = 1}^MA_kx(t ? r_k) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere.     

On the number of types of finite dimensional Hopf algebras

For an odd prime p we construct an infinite class of non-isomorphic Hopf algebras of dimension p ⁴ over an infinite field containing primitive p-th roots of unity, answering in the negative a long standing conjecture of Kaplansky. Oblatum 6-XI-1997 / Published online: 12 November 1998     

On the zeros of certain polynomials

We prove that certain naturally arising polynomials have all of their roots on a vertical line.     

On evaluating multivariate polynomials over finite fields

Some evaluation methods of multivariate polynomials over finite fields are described and their multiplicative complexity is discussed.