David R. Richman

On vector invariants over finite fields

Abstract Let F denote a finite field and let S(m, n, F) denote a set of generators of the invariants of SL(n, F) acting on mn-component vectors. This paper proves that if m > n > 1, then S(m, n, F) must contain a generator whose degree is greater than or equal to (m − n + 2)(|F| − 1). Similar results are obtained for the vector invariants of other groups of matrices with entries in F.

Determination of all imaginary cyclic quartic fields with class number 2

It is proved that there are exactly 8 imaginary cyclic quartic fields with class number 2.

The fundamental theorems of vector invariants

Abstract New proofs are given of the First Fundamental Theorems of vector invariants for the special linear and special orthogonal groups. The novelty of the proofs is that they rely on reducing the theorems to the case of 2 × 2 matrices; there is a close connection between the vector invariants of the special linear group or special orthogonal group of n × n matrices, for all n ⩾ 2, and those of the analogous group of 2 × 2 matrices. This connection is described using the notion of leading monomials. Properties of leading monomials are also used to give new proofs of the Second Fundamental Theorems of vector invariants for the general linear and orthogonal groups.

Calculation of the class numbers of imaginary cyclic quartic fields

Any imaginary cyclic quartic field can be expressed uniquely in the form K = Q(\JA(D + b/d) ), where A is squarefree, odd and negative, D = B2 + C2 is squarefree, B > 0, C > 0, and (A,D)= 1. Explicit formulae for the discriminant and conductor of K are given in terms of A, B, C, D. The calculation of tables of the class numbers h(K) of such fields K is described. Let Q denote the field of rational numbers and let K be a cyclic extension of Q of degree 4. The unique quadratic subfield of K is denoted by k. The class number of K (resp. k) is denoted by h(K) (resp. h(k)). The conductor of K is denoted by / = /( K ). In the case of real cyclic quartic fields K, Gras (3) has given a table of the values of h(K) for all such fields with / < 4000. Recently, the authors have carried out the calculation of the class numbers of imaginary cyclic quartic fields (4). In this note we give a brief description of the computation of the tables given in (4). The following explicit representation of a cyclic quartic field is proved in (4,

On the computation of minimal polynomials

Let K be a field and let f and g be non-constant elements of K[T] . Assume thatgcd(deg f deg g ) is not divisible by the characteristic of K . This paper describes an algorithm to compute the polynomial p(X, Y) of minimal degree such that p(f, g) = 0. Using ideas needed to justify the algorithm, a new proof is given of the fact that if K[f, g]= K[T] , then either deg g dividesdeg f or deg f divides deg g .

A method for determining the reversibility of a Markov sequence

This paper describes, given a tally matrix with strictly positive entries, a method to determine whether the associated Markov process is reversible, and (for reversible Markov processes) methods to compute the reversibility matrix from the tally matrix. If the tally matrixN is symmetric, then it is shown that the Markov process must be reversible and the reversibility matrixC equalss (R−1NR−1), whereR is the diagonal matrix whoseith diagonal entry is the sum of the entries of theith row ofN (for everyi) ands denotes the sum of all the entries ofN. Because a symmetric tally matrix is of special importance in applications, a χ2 test is proposed for determining, in the presence of experimental errors, whether such a matrix is symmetric.

The waring problem for matrices

This paper establishes the following results. Let nand kbe positive integers, with n⩾2. There is a number Gk such that every n×n matrix with integer entries is a sum of Gk k-th powers. Let M(n R) denote the set of n×n matrices with entries in a commutative (and associative) ring R with 1. If n⩾k, the following statements are equivalent: (i) M is a sum of k-th powers in M(n R); (ii) M is a sum of seven k-th powers in M(n R); (iii) Mlies in M(n R) and, for every prime power pe dividing k, there are elements x 0=x 0(p),…,xe =xe (p) in R such that

A Method to Compute Minimal Polynomials

Let

Matrices as sums of squares: a conjecture of Griffin and Krusemeyer

f ( X )

A new proof of a result about Hankel operators

and