We characterize the size of monotone span programs computing certain "structured" boolean functions by the Nullstellensatz degree of a related unsatisfiable Boolean formula.

This yields the first exponential lower bounds for monotone span programs over arbitrary fields, the first exponential separations between monotone span programs over fields of different characteristic, and the first exponential separation between monotone span programs over arbitrary fields and monotone circuits. We also show tight quasipolynomial lower bounds on monotone span programs computing directed st-connectivity over arbitrary fields, separating monotone span programs from non-deterministic logspace and also separating monotone and non-monotone span programs over $GF(2)$. Our results yield the same lower bounds for linear secret sharing schemes due to a known relationship between monotone span programs and linear secret sharing introduced by Karchmer and Wigderson and extended by Beimel.

To prove our characterization we introduce a new and general tool for lifting polynomial degree to rank over arbitrary fields, generalizing a result of Sherstov.