the syzygies of the coordinates of a surface parameterization. Initiated by the geometric modeling community [@SC95], these matrices have been deeply explored in a series of papers (see [@BuJo03; @BC05; @BDD08; @Bot10] and the references therein). Compared to the resultant matrices, they are still very easy to build, although not universal, but their sensitivity to the presence of base points is much weaker. However, these matrices are in general singular matrices and the recovering of the classical implicit equation from them is more involved. Therefore, to be useful these matrices have to be seen as implicit representations on their own, without relying on the more classical implicit equation. In this spirit, the use of these singular matrix representations has recently been explored for addressing the curve/surface and surface/surface intersection problems [@BL12; @LBBM09].

As a continuation, the purpose of this paper is to investigate the self-intersection locus of a surface parameterization through its matrix representations. More precisely, let $\phi$ be a parameterization from ${\mathbb P}^2$ to ${\mathbb P}^3$ of a rational algebraic surface ${\mathcal S}$ and let $M(\phi)$ be one of its matrix representations. The main result of this paper is that the drop of rank of $M(\phi)$ at a given 