On the Ranks and Border Ranks of Symmetric Tensors

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.     

On Octonionic Polynomials

We discuss the generalization of results on quaternionic polynomials to the octonionic polynomials. In contrast to the quaternions the octonionic multiplication is non-associative. This fact although introducing some difficulties nevertheless leads to some new results. For instance, the monic and non-monic polynomials do not have, in general, the same set of zeros. Concerning the zeros, it is shown that in the monic and non-monic cases they are not the same, in general, but they belong to the same set of conjugacy classes. Despite these difficulties created by the non-associativity, we obtain equivalent results to the quaternionic case with respect to the number of zeros and the procedure to compute them.     

On Projective Modules with Constant Ranks

On Projective Modules with Constant RanksIn this paper,we investigate module structures of rings over which every finitely generated projective module with constant rank is stably free. As applications,we give characterizations of some related rings.     

On Ranks of Matrices Associated with Trees

A sign pattern matrix is a matrix whose entries are from the set {+,–,0}. The purpose of this paper is to obtain bounds on the minimum rank of any symmetric sign pattern matrix A whose graph is a tree T (possibly with loops). In the special case when A is nonnegative with positive diagonal and the graph of A is star-like, the exact value of the minimum rank of A is obtained. As a result, it is shown that the gap between the symmetric minimal and maximal ranks can be arbitrarily large for a symmetric tree sign pattern A. Supported by NSF grant No. DMS-00700AMS classification: 05C50, 05C05, 15A48     

On Integer Chebyshev Polynomials

We are concerned with the problem of minimizing the supremum norm on of a nonzero polynomial of degree at most with integer coefficients. We use the structure of such polynomials to derive an efficient algorithm for computing them. We give a table of these polynomials for degree up to and use a value from this table to answer an open problem due to P. Borwein and T. Erdélyi and improve a lower bound due to Flammang et al.

     

On Factorization of Trigonometric Polynomials

We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case, where the outer factorization is also *-outer, is examined in greater detail. The connection to Aglers model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over , in terms of rational functions with fixed denominators.     

On Dirichlet Characters of Polynomials

The main purpose of this paper is to use the properties of Gaussiansums and the analytic method to study the problem of calculatingone kind of character sums modulo q, and to give an interestingidentity. 2000 Mathematics Subject Classification 11L10, 11L40.     

On Ritt's Factorization of Polynomials

Ritt has shown that any complex polynomial p can be writtenas the composition of polynomials p₁,...,p_m, where each p_j isprime in the sense that it cannot be written as a non-trivialcomposition of polynomials. The factors p_j are not unique butthe number m of them is, as is the set of the degrees of thep_j. The paper extends Ritt's theory and, in particular, a thirdinvariant of the decomposition is introduced.     

On Brown Polynomials. II

The results of the author’s previous article are improved. We also establish that each irreducible polynomial over a Henselian valued field is a Brown polynomial.

     

On L p Extremal Polynomials

A Turán-type inequality for L _p extremal polynomials is given and mean convergence of Lagrange interpolation based on the zeros of L _p extremal polynomials is investigated. November 8, 1994. Date revised: January 23, 1997.     

On the Roots of Domination Polynomials

The domination polynomial of a graph G of order n is the polynomial ${D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i}$ where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. We investigate here domination roots, the roots of domination polynomials. We provide an explicit family of graphs for which the domination roots are in the right half-plane. We also determine the limiting curves for domination roots of complete bipartite graphs. Finally, we prove that the closure of the roots is the entire complex plane.     

On the Difference of Orthonormal Polynomials

We establish an estimate on the difference of orthonormal polynomials for a general class of exponential weights.     

某些差分多项式的亏量

Halburd和Korhonen指出研究复域差分的值分布问题对进一步研究复域差分与差分方程具有十分重要的意义.本文得到了关于有限级亚纯函数的差分多项式的亏量为一些结果,其中部分结果可视为微分多项式相应结果的差分模拟.同时,我们在一定条件下给出了经典的Valiron-Mohon'ko定理的一个差分模拟结果,并且作为本文中的一个重要工具出现.这些结果推广了前人已有结果.     

On Idempotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of S is the size of a minimal generating set of S, and the idempotent rank of S is the size of a minimal generating set of S consisting of idempotents in S. A partition of a q-element subset of the set X_n={1,2,..., n} is said to be of type if the sizes of its classes form the partition of q n. A non-trivial partition of a positive integer q consists of k < q elements. For a non-trivial partition of q n, the semigroup S(), generated by all the transformations with kernels of type , is idempotent-generated. It is known that if is a non-trivial partition of n, that is, S() consists of total many-to-one transformations, then the rank and the idempotent rank of S() are both equal to max{ⁿ_d, N()}, where N() is the number of partitions of X_n of type . We extend this result to semigroups of partial transformations, and prove that if is a non-trivial partition of q < n, then the rank and the idempotent rank of S() are both equal to N().     

关于Bernstein多项式的绝对收敛性

该文研究Ｂｅｒｎｓｔｅｉｎ多项式的绝对收敛性．证明了，对每个ｘ∈［０，１］，一个有界变差函数的Ｂｅｒｎｓｔｅｉｎ多项式序列是绝对｜Ｃ，１｜可和的，而且给出了Ｂｅｒｓｔｅｉｎ多项式序列的绝对｜Ｃ，１｜和式的余项的估计．     

On Simple Exponential Sets of Polynomials

In this paper, it is shown that certain classes of regular functions of several complex variables be represented by exponential sets of polynomials in hyperelliptical regions. Moreover, an upper bound for the order of exponential set is given.     

On Second Order Linear Differential Polynomials

We classify all functions φ(z) meromorphic in R ? ¦z¦ < ∞ such that ?(z) Φ(z) has no zeros there, where Φ = (z)′’ + a₁(z) ?′(z) + a₀(z) ?(z) and a₀(z), a₁(z) are rational at infinity.