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We study the primitive recursive content of various closure results in algebra and model theory, including the algebraic, the real, and the differential closure theorems. In the case of ordered fields and their real closures, our result settles a question recently raised by Selivanova and Selivanov. [ABSTRACT FROM AUTHOR]

We give a criterion when a polynomial x n − g is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic. [ABSTRACT FROM AUTHOR]

Inspired by a recent paper of Alain Connes and Caterina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces (“Singer actions”), we consider several fundamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields F the plane P 2 ( F ) = PG ( 2 , F ) (and more generally the space P n ( F ) = PG ( n , F ) ) admits a Singer group, and show, e.g., that for any prime p and any positive integer n > 1 , PG ( n , F p ‾ ) cannot admit Singer groups ( F p ‾ an algebraic closure of F p ). One of the main results in characteristic 0, which is a corollary of a criterion which applies to many other fields, is that PG ( m , R ) with m ≠ 0 a positive even integer, cannot admit Singer groups. [ABSTRACT FROM AUTHOR]

Algebraically closed abelian groups are precisely the divisible abelian groups; the underlying groups of rational vector spaces are precisely the divisible abelian groups. That is, rational vector spaces are the skolemization of algebraically closed abelian groups, obtained by adding scalar multiplication by each rational to the set of operations. Algebraically closed bounded distributive lattices are precisely the complemented distributive lattices; the underlying lattices of boolean algebras are precisely the complemented distributive lattices. That is, boolean algebras are the skolemization of algebraically closed bounded distributive lattices, obtained by adding complementation to the set of operations. We explore this idea for arbitrary universal classes of algebras and focus particularly on the case of meet semilattices. The algebraically closed meet semilattices are precisely the distributive meet semilattices; their skolemization seems not to have been discussed previously in the literature. We discuss it here but reach no firm solution. [ABSTRACT FROM AUTHOR]

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]

Let A be an algebra (of an arbitrary finitary type), and let γ be a binary term. A pair (a, b) of elements of A will be called a γ- eligible pair if for each x in the subalgebra generated by {a, b} such that x is distinct from a there exists an element y in A such that b = xyγ. We say that A is a γ- closed algebra if for each γ-eligible pair (a, b) there is an element c with b = acγ. We call A a closed algebra if it is γ-closed for all binary terms γ that do not induce a projection. Let T be a unital subring of the field of real numbers. Equipped with all the binary operations $${(x, y) \mapsto (1- p)x+py}$$ for $${p \in T}$$ and 0 < p < 1, T becomes a mode, that is, an idempotent algebra in which any two term functions commute. In fact, the mode T is a (generalized) barycentric algebra. Let $${\mathcal{Q}(T)}$$ denote the quasivariety generated by this mode. Our main theorem asserts that each mode of $${\mathcal{Q}(T)}$$ extends to a minimal closed cancellative mode, which is unique in a reasonable sense. In fact, we prove a slightly stronger statement. As corollaries, we obtain a purely algebraic description of the usual topological closure of convex sets, and we exemplify how to use the main theorem to show that certain open convex sets are not isomorphic. [ABSTRACT FROM AUTHOR]

In this paper, we extend the definition of the Nathanson height from points in projective spaces over to points in projective spaces over arbitrary finite fields. If , then the Nathanson height is where with the field norm and the element of congruent to modulo p. We investigate the basic properties of this extended height, provide some bounds, study its image on the projective line and propose some questions for further research. [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]

Abstract: The problem to express an matrix A as the sum of two square-zero matrices was first investigated by Wang and Wu for matrices over the complex field. This paper investigates the problem over an arbitrary field F. It is shown that, if char, then is the sum of two square-zero matrices if and only if A is similar to a matrix of the form , where N is nilpotent, X is nonsingular, and each is a companion matrix associated with an even-power poly nomial with nonzero constant term. If F is of characteristic two, the term falls away. If F is of characteristic zero and algebraically closed, the term falls away and the result of Wang and Wu is obtained. [Copyright &y& Elsevier]

Characteristic matrices and metrics of equivalence systems are studied that help give an efficient description of conjunctions of equivalence systems. Using these results, families of correct polynomials in the algebraic approach to classification are described. [ABSTRACT FROM AUTHOR]

Abstract: According to Markov (1946) , a subset of an abelian group G of the form , for some integer n and some element , is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally bounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian topology on G called the Zariski, or verbal, topology of G; see Bryant (1977) . We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Fréchet–Urysohn. For a countable family of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology on G such that the -closure of each member of coincides with its -closure. As an application, we provide a characterization of the subsets of G that are -dense in some Hausdorff group topology on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov (1946) . [ABSTRACT FROM AUTHOR]

Abstract: A practical computational system is described for computing with an algebraic closure of a field. The system avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. All roots of an arbitrary polynomial defined over such an algebraically closed field can be constructed and are easily distinguished within the system. The difficult case of inseparable extensions of function fields of positive characteristic is also handled properly by the system. A technique of modular evaluation into a finite field critically ensures that a unique genuine field is simulated by the system but also provides fast optimizations for some fundamental operations. Fast matrix techniques are also used for several non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation. [Copyright &y& Elsevier]

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points. [ABSTRACT FROM AUTHOR]

We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil–Rödl Theorem, the Ramsey property of partial orders and metric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered S -metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the structural Ramsey theorem. Both of these results are natural, and easy to state, yet their proofs involve most of the theory developed here. We also characterise Ramsey lifts of classes of structures defined by finitely many forbidden homomorphisms and extend this to special cases of classes with closures. This has numerous applications. For example, we find Ramsey lifts of many Cherlin–Shelah–Shi classes. [ABSTRACT FROM AUTHOR]