Some quantitative results in singularity theory
The classical singularity theory deals with singularities of various mathematical objects: curves and surfaces, mappings, solutions of differential equations, etc. In particular, singularity theory treats the tasks of recognition, description and classification of singularities in each of these cases. In many applications of singularity theory it is important to sharpen its basic results, making them “quantitative”, i.e. providing explicit and effectively computable estimates for all the important parameters involved. This opens new possibilities for applications in analysis, geometry, differential equations, dynamics, and, last not least, in computations. Application of the results of singularity theory in numerical data processing with finite accuracy stresses another important requirement: the “normalizing transformations” must be explicitly computable. The most natural interpretation of this requirement is in terms of the “jet calculus”: given the Taylor polynomials of the input data, we should be able to produce explicitly the Taylor polynomials of the output normalizing transformations. This papers provides a sample of initial results in these directions.