Using Smith normal forms and μ-bases to compute all the singularities of rational planar curves

Abstract

We prove a result similar to the conjecture of Chen et al. (2008) concerning how to calculate the parameter values corresponding to all the singularities, including the infinitely near singularities, of rational planar curves from the Smith normal forms of certain Bezout resultant matrices derived from μ-bases. A great deal of mathematical lore is hidden behind their conjecture, involving not only the classical blow-up theory of singularities from algebraic geometry, but also the intrinsic relationship between μ-bases and the singularities of rational planar curves. Here we explore these mathematical foundations in order to reveal the true nature of this conjecture. We then provide a novel approach to proving a related conjecture, which in addition to these mathematical underpinnings requires only an elementary knowledge of classical resultants.