Using character varieties: Presentations, invariants, divisibility and determinants

If G is a finitely generated group, then the set of all characters from G into a linear algebraic group is a useful (but not complete) invariant of G . In this thesis, we present some new methods for computing with the variety of SL2C -characters of a finitely presented group. We review the theory of Fricke characters, and introduce a notion of presentation simplicity which uses these results. With this definition, we give a set of GAP routines which facilitate the simplification of group presentations. We provide an explicit canonical basis for an invariant ring associated with a symmetrically presented group\u27s character variety. Then, turning to the divisibility properties of trace polynomials, we examine a sequence of polynomials rn(a) governing the weak divisibility of a family of shifted linear recurrence sequences. We prove a discriminant/determinant identity about certain factors of rn( a) in an intriguing manner. Finally, we indicate how ordinary generating functions may be used to discover linear factors of sequences of discriminants. Other novelties include an unusual binomial identity, which we use to prove a well-known formula for traces; the use of a generating function to find the inverse of a map xn ∣→ fn(x); and a brief exploration of the relationship between finding the determinants of a parametrized family of matrices and the Smith Normal Forms of the sequence

La queste del saint Gra(AL): A computational approach to local algebra

AbstractWe show how, by means of the Tangent Cone Algorithm, the basic functions related to the maximal ideal topology of a local ring can be effectively computed in the situations of geometrical significance, i.e.:(1)localizations of coordinate rings of a variety at the prime ideal defining a subvariety,(2)rings of algebraic formal power series rings.In particular we show how the method of “associated graded rings” can be turned into an effective tool to compute local algebraic invariants of varieties

Lazy exact real computation

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Linear Codes Defined From Higher -Dimensional Varieties.

We establish an algebraic foundation to complement the improved geometric codes of Feng and Rao. Viewing linear codes as affine variety codes, we utilize the Feng-Rao minimum distance bound to construct codes with relatively large dimensions. We examine higher-dimensional affine hypersurfaces with properties similar to those of Hermitian curves. We determine a Grobner basis for the ideal of the variety of rational points on certain affine Fermat varieties. This result is applied to determine parameters of codes defined from Fermat surfaces

The Bernstein basis in set-theoretic geometric modelling

SIGLEAvailable from British Library Document Supply Centre-DSC:DXN037062 / BLDSC - British Library Document Supply CentreGBUnited Kingdo