Resultants,companion matrices,and Jacobson chains

The multiplicative property of triangular Toeplitz matrices is used to derive a polyproduct rule for companion matrices. This rule is then used to obtain some new identities involving resultants and companion matrices, and to investigate the block structure of Jacobson chains relating companion and hypercompanion matrices.     

Resultants and contour integrals

Resultants are important special functions used to describe nonlinear phenomena. The resultant R_{r1 ?rn} R_{r_1 \ldots r_n } determines a consistency condition for a system of n homogeneous polynomials of degrees r ₁, ..., r _n in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications.     

Information about group matrices

Each ordering for the elements of a finite group G of order n defines a corresponding class of group matrices for G. First, this paper proves that the number of distinct classes of group matrices for G equals (n ? 1)!/m, where m is the number of automorphisms of G. Then, a study is made of a block-diagonal reduction for the group matrices of any particular class.     

Oblique projectors and group involutory matrices

In this work, conditions characterizing when a lineal combination of two projectors of the same order is a group involutory matrix are analyzed. In particular, a representation of an involutory matrix as a sum or difference of two projectors is given. In addition, some results are applied to control problems. An application to obtain a state feedback control is given.     

On the group matrices for a generalized dihedral group

For the generalized dihedral group, an effective method is given to construct a matrix which transforms group matrices to a convenient block-diagonal form.     

Free group of infinite unitriangular matrices

In the group of infinite unitriangular matrices over the field with two elements, a free subgroup of rank two is constructed which is a group of finite-automata transformations over a two-element alphabet. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 382–386, March, 2000.     

Factorization and Resultants of Partial Differential Operators

Comparatively little is known about commutative rings of partial differential operators, while in the ordinary case, concrete examples and an algebraic(-geometric) structure can be algorithmically determined for large classes. In this note, by the calculation of the partial μ-shifted differential resultant which we defined in a previous paper, we produce algebraic equations of spectral surfaces for commutative rings in two variables, and Darboux transformations of Airy-type operators that correspond to morphisms of surfaces. There are, however, many elementary differential-algebraic statements that we only observe experimentally, thus we offer open questions which seem to us quite significant in differential algebra, and access to Mathematica code to enable further experimentation.     

Resultants and Chow forms via exterior syzygies

Given a sheaf on a projective space , we define a sequence of canonical and effectively computable Chow complexes on the Grassmannians of planes in , generalizing the well-known Beilinson monad on . If the sheaf has dimension , then the Chow form of the associated -cycle is the determinant of the Chow complex on the Grassmannian of planes of codimension . Using the theory of vector bundles and the canonical nature of the complexes, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford bundle gives rise to a polynomial formula for the resultant of five homogeneous forms of degree eight in five variables.

     

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

On nonsingularity of combinations of three group invertible matrices and three tripotent matrices

Let T?=?c ₁ T ₁?+?c ₂ T ₂?+?c ₃ T ₃???c ₄(T ₁ T ₂?+?T ₃ T ₁?+?T ₂ T ₃), where T ₁, T ₂, T ₃ are three n?×?n tripotent matrices and c ₁, c ₂, c ₃, c ₄ are complex numbers with c ₁, c ₂, c ₃ nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices.     

On nonsingularity of combinations of two group invertible matrices and two tripotent matrices

Let T ₁ and T ₂ be two n?×?n tripotent matrices and c ₁, c ₂ two nonzero complex numbers. We mainly study the nonsingularity of combinations T?=?c ₁ T ₁?+?c ₂ T ₂???c ₃ T ₁ T ₂ of two tripotent matrices T ₁ and T ₂, and give some formulae for the inverse of c ₁ T ₁?+?c ₂ T ₂???c ₃ T ₁ T ₂ under some conditions. Some of these results are given in terms of group invertible matrices.     

Resultants and discriminants of Chebyshev and related polynomials

We show that the resultants with respect to of certain linear forms in Chebyshev polynomials with argument are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.

     

Space Saving Calculation of Symbolic Resultants

We describe an approach to the computation of symbolic resultants in which factors are removed during the course of the calculation, so reducing the stack size required for intermediate expressions and the storage space needed. We apply the technique to three well-established methods for calculating resultants. We demonstrate the advantages of our approach when the resultants are large and show that some otherwise intractable problems can be resolved. In certain cases a significant reduction in the cpu time required to calculate the resultant is also evident.      

Finite groups of matrices over group rings

We investigate certain finite subgroups of , where is a finite nilpotent group. Such a group gives rise to a -module; we study the characters of these modules to limit the structure of . We also exhibit some exotic subgroups .