Matrix model curve near the singularities

In [Formula Presented] supersymmetric [Formula Presented] gauge theories with the tree-level superpotential [Formula Presented] that is an arbitrary polynomial of the adjoint matter [Formula Presented] the massless fluctuations about each quantum vacuum are described by [Formula Presented] gauge theory. By turning on the parameters of [Formula Presented] to the special values, the singular vacua where the additional fields become massless can be reached. Using the matrix model prescription, we study the intersections of [Formula Presented] and [Formula Presented] branches. The general formula for the matrix model curve at the singularity which is valid for arbitrary N is obtained and this generalizes the previous results for small values of N from the strong-coupling approach. Applying the analysis to the degenerated case, we also obtain a general matrix model curve which is not only valid at a special point but also on the whole branch.