Euclidean Quadratic Forms and ADC Forms I

Motivated by classical results of Aubry, Davenport and Cassels, we define the notion of a Euclidean quadratic form over a normed integral domain and an ADC form over an integral domain. The aforementioned classical results generalize to: Euclidean forms are ADC forms. We then initiate the study and classification of these two classes of quadratic forms, especially over discrete valuation rings and Hasse domains.Comment: 26 page

Covering Numbers in Linear Algebra

We compute the minimal cardinality of a covering (resp. an irredundant covering) of a vector space over an arbitrary field by proper linear subspaces. Analogues for affine linear subspaces are also given.Comment: 4 page