Calculating Tangent Cones With Triangular Decomposition.
Paul Vrbik, Western
12:30pm, Wednesday June 4th, K9509.
Abstract: Sometimes it is useful to investigate the local behaviour of a hypersurface about a point. When this point is ‘nice’ (which in our case means non-singular) the tangent plane provides an adequate linear description of the space. However, when this point is singular, a bundle of tangens lines is required. This bundle is called the Tangent Cone and when working algebraically one can calculate this by finding the homogeneous components of least degree among the ideal the hypersurface generates. This is easy for principally generated ideals, but for ideals with more than one generator Standard bases are required which in turn require Groebner bases (GBs). As these GBs are often prohibitively expensive to calculate there is utility in finding a GB-free algorithm. Indeed one can take an "algebraic limit" by finding secants modulo triangular sets (in particular regular chains) utilizing efficient splitting algorithms. We present an algorithm for computing a tangent cone of a one-dimensional ideal at a point given by a zero-dimensional ideal using Triangularization.