A codimension formula for intrinsic perturbation of $A$-hypergeometric series

We study the canonical formal solution space of a homogeneous $A$-hypergeometric system in a generic direction. We present as a finite-dimensional cokernel the quotient of that space by the span of the canonical series obtained by intrinsic perturbation, taken over all exponents occurring in that space and all ordered negative support families, and we compute the codimension of that span. We show that the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. We apply these results, for every integer $q\geq 1$, to a homogeneous $A$-hypergeometric system of lattice rank 2 with $5q$ columns and show that the span of the canonical series obtained by intrinsic perturbation has codimension at least $\lceil 3q^2/4\rceil$ in the canonical formal solution space.