Khovanskii's Bezout-type Theorem for Pfaffian Functions: A Self-Contained Proof, and Applications

What the math gives to ML

The paper provides an explicit upper bound on the number of regular real zeros of systems sharing a Pfaffian chain, refined by the number of input variables actually used by the chain rather than the ambient dimension. This can become a structural complexity budget for neural networks whose activations satisfy low-degree differential equations, such as tanh. The practical transfer is to constrain hidden width, activation-chain complexity, and input-group sparsity, then test whether the resulting bound correlates with fewer oscillatory decision-boundary crossings and better robustness. The theorem is too conservative to serve as a directly optimized loss, so it is best used for architecture selection and as a validation certificate.