Notation and Conventions

• Rn denotes the Euclidean space of dimension n. Subsets of Rn are viewed as metric spaces using the standard Euclidean distance on Rn. • Mn(R) denotes the real vector space of n× n real matrices with the Euclidean metric, and I denotes the identity matrix in Mn(R). • All rings are associative, with a multiplicative identity. • For any prime number p, Fp denotes the finite field with p elements. • If A and B are sets, then A−B refers to {x ∈ A | x 6∈ B}. • For a ring R, R[x] denotes the polynomial ring in one variable over R, and R[x, y] denotes the polynomial ring in two variables over R.