Peter Vámos

Length functions, multiplicities and algebraic entropy

Abstract. We consider algebraic entropy defined using a general discrete length function L and will consider the resulting entropy in the setting of -modules. Then entropy will be viewed as a function on modules over the polynomial ring extending L. In this framework we obtain the main results of this paper, namely that under some mild conditions the induced entropy is additive, thus entropy becomes an operator from the length functions on R-modules to length functions on -modules. Furthermore, if one requires that the induced length function satisfies two very natural conditions, then this process is uniquely determined. When R is Noetherian, we will deduce that, in this setting, entropy coincides with the multiplicity symbol as conjectured by the second named author. As an application we show that if R is also commutative, the L-entropy of the right Bernoulli shift on the infinite direct product of a module of finite positive length has value , generalizing a result proved for Abelian groups by A. Giordano Bruno.

Block diagonalization and 2-unit sums of matrices over Prüfer domains

We show that matrices over a large class of Prüfer domains are equivalent to “almost diagonal” matrices, that is, to matrices with all the nonzero entries congregated in blocks along the diagonal, where both dimensions of the diagonal blocks are bounded by the size of the class group of the Prüfer domain. This result, a generalization of a 1972 result of L. S. Levy for Dedekind domains, implies that, for n sufficiently large, every n× n matrix is a sum of two invertible matrices. We also generalize from Dedekind to certain Prüfer domains a number of results concerning the presentation of modules and the equivalence of matrices presenting them, and we uncover some connections to combinatorics.

An algorithm to compute the set of characteristics of a system of polynomial equations over the integers

We describe a (finite) algorithm to determine the set of characteristics of a system of polynomial equations with integer coefficients by using the theory of Grobner bases. This gives us a proof that the set of characteristics must be either finite and not containing zero, or containing zero and co-finite. Another, algebraic, proof of this is given in the appendix. These results carry over to systems of polynomial equations over a principal ideal domain and also yields an algorithm for finding the characteristic set of a matroid.

The Hopfian exponent of an abelian group

If

2-GOOD RINGS

G

The dimension of the tensor product of a finite number of field extensions

G is a Hopfian abelian group then it is, in general, difficult to determine if direct sums of copies of

A note on clean abelian groups

G