The holonomic rank of a holonomic system is the dimension of its space of local holomorphic solutions at a generic nonsingular point (Saito et al. 2000). It plays the role for a system of linear partial differential equations that the order plays for an ordinary differential equation, counting the local degrees of freedom in a solution. When this system is rewritten as a first-order Pfaffian system using a basis of independent derivatives, the solution vector has length equal to the rank. The connection matrices express the partial derivatives of this vector in the chosen basis and are square matrices of the same size.
For a Horn system, the rank counts all locally independent solution branches, not merely the distinguished Horn-series solution (Dickenstein et al. 2005). It is also a practical measure of the computational complexity of numerical evaluation, since it controls the size of the Pfaffian system used in numerical continuation (Banik and Bera 2026).
