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We consider the large $N$ expansion of the partition function for theHermitian one-matrix model. It is well known that the coefficients of thisexpansion are generating functions $F^{(g)}$ for a certain kind of graphembedded in a Riemann surface. Other authors have made a simplifying assumptionthat 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 onthe string equations, and yields "valence independent" formulas which do notdepend explicitly on the potential. We introduce a family of differentialoperators, the "string polynomials", which make clear the valence independentnature of the string equations.

Solution of the string equations for asymmetric potentials