We explore an arithmetic analogue of the numerical range. We define the numerical range of a square matrix with entries in a finite field Zp[i], for any prime p congruent to 3 modulo 4. We establish the basic properties of these new numerical ranges, and prove several foundational results for matrices of arbitrary dimension. We classify the shapes of the numerical ranges of 2 × 2 matrices over these finite fields.

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In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of n × n Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is 2n − 3.

Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the existence of maximum likelihood estimates or the normality of the associated semigroup. Cut polytopes of graphs have been useful in analyzing binary marginal polytopes in the case where the simplicial complex underlying the hierarchical model is a graph. We introduce a generalized cut polytope that is isomorphic to the binary marginal polytope of an arbitrary simplicial complex via a generalized covariance map. This polytope is full dimensional in its ambient space and has a natural switching operation among its facets that can be used to deduce symmetries between the facets of the correlation and binary marginal polytopes. We find complete H-representations of the generalized cut polytope for some important families of simplicial complexes. We also compute the volume of these polytopes in some instances.