The expected number of zeros of a random system of \(p\)-adic polynomials. (English) Zbl 1130.60010

Evans, Steven N.

Electron. Commun. Probab. 11, 278-290 (2006).

Summary: 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 \(p^{d \lfloor \log\_{p} N \rfloor} (1 + p^{-1} + p^{-2} + \cdots + p^{-d})^{-1}\) for the simplest such model.

MSC:

60B99	Probability theory on algebraic and topological structures
11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
30G06	Non-Archimedean function theory