The expected number of zeros of a random system of p-adic polynomials

We study the simultaneous zeros of a random family of d polynomials in d variables over the p-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the d-fold Cartesian product of the p-adic integers. Considering models in which the maximum degree that each variable appears is N, this expected value is pd logₚ N (1 + p⁻¹ + p⁻² + ⋯ + p⁻ᵈ)⁻¹ for the simplest such model.