Partitions of groups and matroids into independent subsets

Can the real line with removed zero be covered by countably many linearly (algebraically) independent subsets over the field of rationals? We use a matroid approach to show that an answer is "Yes" under the Continuum Hypothesis, and "No" under its negation.Comment: 5 page

The Alternating Stock Size Problem and the Gasoline Puzzle

Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that: (i) all prefix sums of the ordering are nonnegative, and (ii) the maximum value of a prefix sum is minimized. Kellerer et al. referred to this problem as the "Stock Size Problem" and showed that it can be approximated to within 3/2. They also showed that an approximation ratio of 2 can be achieved via several simple algorithms. We consider a related problem, which we call the "Alternating Stock Size Problem", where the number of positive and negative integers in the input set S are equal. The problem is the same as above, but we are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations. We show that it can be approximated to within 1.79. Then we show that this problem is closely related to an optimization version of the gasoline puzzle due to Lov\'asz, in which we want to minimize the size of the gas tank necessary to go around the track. We present a 2-approximation for this problem, using a natural linear programming relaxation whose feasible solutions are doubly stochastic matrices. Our novel rounding algorithm is based on a transformation that yields another doubly stochastic matrix with special properties, from which we can extract a suitable permutation

Computing generators of free modules over orders in group algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition of E[G] is explicitly computable and each component is in fact a matrix ring over a field, this leads to an algorithm that either gives an A-basis for X or determines that no such basis exists. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A of O_L in E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K=E=Q.Comment: 17 pages, latex, minor revision

The real field with an irrational power function and a dense multiplicative subgroup

This paper provides a first example of a model theoretically well behaved structure consisting of a proper o-minimal expansion of the real field and a dense multiplicative subgroup of finite rank. Under certain Schanuel conditions, a quantifier elimination result will be shown for the real field with an irrational power function and a dense multiplicative subgroup of finite rank whose elements are algebraic over the field generated by the irrational power. Moreover, every open set definable in this structure is already definable in the reduct given by just the real field and the irrational power function