Solution of the string equations for asymmetric potentials

We consider the large $N$ expansion of the partition function for the Hermitian one-matrix model. It is well known that the coefficients of this expansion are generating functions $F^{(g)}$ for a certain kind of graph embedded in a Riemann surface. Other authors have made a simplifying assumption that the potential $V$ is an even function. We present a method for computing $F^{(g)}$ in the case that $V$ is not an even function. Our method is based on the string equations, and yields "valence independent" formulas which do not depend explicitly on the potential. We introduce a family of differential operators, the "string polynomials", which make clear the valence independent nature of the string equations.