On the Arithmetic of Function Fields

[[abstract]]這篇論文包含三個章節, 都是討探在係數為有限體的多項環上的算術問題. 此三個問題分別為: On the Polynomial Gauss Sums(mod P^n),n>1, On the Polynomial Schur's Matrix, 和 On the Number of Solutions of Linear Equation in Finite Carlitz Modules.

Prehomogeneous vector spaces and ergodic theory III

Let H_1=SL(5), H_2=SL(3), H=H_1 \times H_2. It is known that (G,V) is a prehomogeneous vector space (see [22], [26], [25], for the definition of prehomogeneous vector spaces). A non-constant polynomial \delta(x) on V is called a relative invariant polynomial if there exists a character \chi such that \delta(gx)=\chi(g)\delta(x). Such \delta(x) exists for our case and is essentially unique. So we define V^{ss}={x in V such that \delta(x) is not equal to 0}. For x in V_R^{ss}, let H_{x R+}^0 be the connected component of 1 in classical topology of the stabilizer H_{x R}. We will prove that if x in V_R^ss is "sufficiently irrational", H_{x R+}^0 H_Z is dense in H_R

THE ALGEBRA GENERATED BY THREE COMMUTING MATRICES

Abstract. We present a survey of an open problem concerning the dimension of the algebra generated by three commuting matrices. This article concerns a problem in algebra that is completely elementary to state, yet, has proven tantalizingly difficult and is as yet unsolved. Consider C[A, B, C] , the C-subalgebra of the n × n matrices Mn(C) generated by three commuting matrices A, B, and C. Thus, C[A, B, C] consists of all C-linear combinations of “monomials ” A i B j C k, where i, j, and k range from 0 to infinity. Note that C[A, B, C] and Mn(C) are naturally vector-spaces over C; moreover, C[A, B, C] is a subspace of Mn(C). The problem, quite simply, is this: Is the dimension of C[A, B, C] as a C vector space bounded above by n? Note that the dimension of C[A, B, C] is at most n 2, because the dimension of Mn(C) is n 2. Asking for the dimension of C[A, B, C] to be bounded above by n when A, B, and C commute is to put considerable restrictions on C[A, B, C]: this is to require that C[A, B, C] occupy only a small portion of the ambient Mn(C) space in which it sits. Actually, the dimension of C[A, B, C] is already bounded above by something slightly smaller than n 2, thanks to a classical theorem of Schur ([16]), who showed that the maximum possible dimension of a commutative C-subalgebra of Mn(C) is 1 + ⌊n2 /4⌋. But n is small relative even to this number. To understand the interest in n being an upper bound for the dimension of C[A, B, C], let us look more generally at the dimension of the C-subalgebra of Mn(C) generated by k-commuting matrices. Let us start with the k = 1 case: note that “one commuting matrix ” is just an arbitrary matrix A. Recall that the Cayley-Hamilton theorem tells us that A n is a linear combination of I, A,..., A n−1, where I stands for the identity matrix. From this, it follows by repeated reduction that A n+1, A n+2, etc. are all linear combinations of I, A,..., A n−1 Thus, C[A], the C-subalgebra of Mn(C) generated by A, is of dimension at most n, and this is just a simple consequence of Cayley-Hamilton. The case k = 2 is therefore the first significant case. It was treated by Gerstenhaber ([4]) as well as Motzkin and Taussky-Todd ([13], who proved independently that the variety of commuting pairs of matrices is irreducible. It follows from this that if A and B are two commuting matrices, then too

Permutation polynomials and systems of permutation polynomials in several variables over finite rings

This paper will present the historical development of theorems regarding permutation polynomials in several variables over finite fields. Single variable permutation polynomials will be discussed since they are so important to the discussions which will follow. Theorems involving permutation polynomials and systems of permutation polynomials will also be considered. It will be shown that many of the interesting results obtained for finite fields can be generalized to finite rings

On a Frobenius Problem for Polynomials

We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~k. Similar to the classical problem we show that the n = 2 case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over ℤ to k[t] and give an algorithm to solve the Frobenius problem for polynomials over a field k of sufficiently large size

A constant term identity featuring the ubiquitous (and mysterious) Andrews-Mills-Robbins-Rumsey numbers 1, 2, 7, 42, 429, …

AbstractAndrews's recent proof of the Mills-Robbins-Rumsey conjectured formula for the number of totally symmetric self-complementary plane partitions is used to derive a new multi-variate constant term identity, reminiscent of, but not implied by, Macdonald's BCn-Dyson identity. The method of proof consists in translating to the language of constant terms an expression of Doran for the desired number in terms of sums of minors of a certain matrix. The question of a direct proof of the identity, which would furnish an alternative proof of the Mills-Robbins-Rumsey conjecture, is raised, and a prize is offered for its solution