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Convex subsets of affine spaces over the field of real numbers are described by so-called barycentric algebras. In this paper, we discuss extensions of the geometric and algebraic definitions of a convex set to the case of more general coefficient rings. In particular, we show that the principal ideal subdomains of the reals provide a good framework for such a generalization. Since the closed intervals of these subdomains play an essential role, we provide a detailed analysis of certain cases, and discuss differences from the "classical" intervals of the reals. We introduce a new concept of an algebraic closure of "geometric" convex subsets of affine spaces over the subdomains in question, and investigate their properties. We show that this closure provides a purely algebraic description of topological closures of geometric generalized convex sets. Our closure corresponds to one instance of the very general closure introduced in an earlier paper of the authors. The approach used in this paper allows to extend some results from that paper. Moreover, it provides a very simple description of the closure, with concise proofs of existence and uniqueness. [ABSTRACT FROM AUTHOR]

Given an irreducible polynomial f in k[X1,..., Xn] (where k is a field) such that k is algebraically closed in the quotient field of A ≔ k[X1,...,Xn]/f k[X1,...,Xn], we show that k(f) is algebraically closed in k(X1,...,Xn). Further, if n ≥ 2 and char k = 0, then we show that the number of k-translates of f that are reducible in k[X1,..., Xn] is bounded above by the rank of U(A)/U(k). Finally, we prove a similar bound for the number of reducible composites of the form Γ(f) with Γ ∈ k[T] monic irreducible. [ABSTRACT FROM AUTHOR]