A description of <Emphasis Type="Italic">n</Emphasis>-ary semigroups polynomial-derived from integral domains

We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Głazek and Gleichgewicht’s classification of the corresponding ternary semigroups.     

Algorithm of polynomial complexity for factoring polynomials over local fields

An algorithm is obtained for factoring polynomials in several variables over local fields with complexity which is polynomial in the length of notation of the input data and the characteristic of the residue field of the local field. Here by definition we assume that an infinite series can be calculated in polynomial time if its i-th partial sum can be calculated in time which is polynomial in the length of notation of the input data and i for any i.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 112–148, 1991.     

Conditions on decomposable symmetric tensors as an algebraic variety

Let V be a finite dimensional vector space of dimension at least 2 over an infinite field F. We show that the set of all decomposable elements in the rth symmetric product space over i:V(r≥ 2) is an algebraic set if F is algebraically closed and only if every polynomial of degree at most r splits completcly over F.     

On tree‐decompositions of one‐ended graphs

A graph is one‐ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex v dominates a ray in the end if there are infinitely many paths connecting v to the ray such that any two of these paths have only the vertex v in common. We prove that if a one‐ended graph contains no ray which is dominated by a vertex and no infinite family of pairwise disjoint rays, then it has a tree‐decomposition such that the decomposition tree is one‐ended and the tree‐decomposition is invariant under the group of automorphisms. This can be applied to prove a conjecture of Halin from 2000 that the automorphism group of such a graph cannot be countably infinite and solves a recent problem of Boutin and Imrich. Furthermore, it implies that every transitive one‐ended graph contains an infinite family of pairwise disjoint rays.     

Sur les algèbres absolument valuées qui vérifient l'identité (x, x, x) = 0

Let A denote a prehilbert absolute valued real algebra such that (x, x, x) = 0 for all x ε A; for this algebra we obtain the same results we have previously obtained for the flexible absolute valued algebra. Our main theorem is: A has a finite dimension 1, 2, 4 or 8, and is isotopic to or C. One of the results concerning the isomorphism between A and , C^*, or C shows that if for every two idempotents e₁ and e₂ in , then A is isomorphic to , C^*, or C. The example of infinite dimensional Hilbert absolute valued algebra given by Urbanik and Wright indicates that the assumption, (x, x, x) = 0 for all x ε A, is essential.     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

Partial Skew Polynomial Rings and Jacobson Rings

In this article we consider rings R with a partial action α of an infinite cyclic group G on R. We generalize the well-known results about Jacobson rings and strongly Jacobson rings in skew polynomial rings and skew Laurent polynomial rings to partial skew polynomial rings and partial skew Laurent polynomial rings.     

Non‐discrete metrics in and some notions of finiteness

We show that (i) it is consistent with that there are infinite sets X on which every metric is discrete; (ii) the notion of real infinite is strictly stronger than that of metrically infinite; (iii) a set X is metrically infinite if and only if it is weakly Dedekind‐infinite if and only if the cardinality of the set of all metrically finite subsets of X is strictly less than the size of ; and (iv) an infinite set X is weakly Dedekind‐infinite if and only if has infinite towers if and only if X has countable partitions.     

One‐way infinite 2‐walks in planar graphs

We prove that every 3‐connected 2‐indivisible infinite planar graph has a 1‐way infinite 2‐walk. (A graph is 2‐indivisible if deleting finitely many vertices leaves at most one infinite component, and a 2‐walk is a spanning walk using every vertex at most twice.) This improves a result of Timar, which assumed local finiteness. Our proofs use Tutte subgraphs, and allow us to also provide other results when the graph is bipartite or an infinite analog of a triangulation: then the prism over the graph has a spanning 1‐way infinite path.     

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.     

Profinite groups with nonabelian crowns of bounded rank and their probabilistic zeta function

It has been conjectured by Mann that the infinite sum Σ _H μ(H,G)/|G:H| ^s , where H ranges over all open subgroups of a finitely generated profinite group G, converges absolutely in some half right plane if G is positively finitely generated. We prove that the conjecture is true if the nonabelian crowns of G have bounded rank. In particular Mann’s conjecture holds if G has polynomial subgroup growth or is an adelic profinite group.     

Generalized Poincaré Classes and Cubic Equivalences

Let Y be a smooth projective algebraic surface over ?, and T(Y) the kernel of the Albanese map CH₀(Y)_deg0 → Alb(Y). It was first proven by D. Mumford that if the genus Pg(Y) > 0, then T(Y) is 'infinite dimensional'. One would like to have a better idea about the structure of T(Y). For example, if Y is dominated by a product of curves E₁ × E₂, such as an abelian or a Kummer surface, then one can easily construct an abelian variety B and a surjective 'regular' homomorphism B^?z2 → T(Y). A similar story holds for the case where Y is the Fano surface of lines on a smooth cubic hypersurface in P⁴. This implies a sort of boundedness result for T(Y). It is natural to ask if this is the case for any smooth projective algebraic surface Y ? Partial results have been attained in this direction by the author [Illinois. J. Math. 35 (2), 1991]. In this paper, we show that the answer to this question is in general no. Furthermore, we generalize this question to the case of the Chow group of k—cycles on any projective algebraic manifold X, and arrive at, from a conjectural standpoint, necessary and sufficient cohomological conditions on X for which the question can be answered affirmatively.     

Partial Skew Polynomial Rings and Goldie Rings

We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R?x; α? is right Goldie, where R[x; α] (R?x; α?) denotes the partial skew (Laurent) polynomial ring over R. In addition, R?x; α? is semiprime while R[x; α] is not necessarily semiprime.     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tetravalent vertex‐transitive graphs of order 4p

A graph is vertex‐transitive if its automorphism group acts transitively on vertices of the graph. A vertex‐transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this article, the tetravalent vertex‐transitive non‐Cayley graphs of order 4p are classified for each prime p. As a result, there are one sporadic and five infinite families of such graphs, of which the sporadic one has order 20, and one infinite family exists for every prime p>3, two families exist if and only if p≡1 (mod 8) and the other two families exist if and only if p≡1 (mod 4). For each family there is a unique graph for a given order. © 2011 Wiley Periodicals, Inc.     

Compléments à un Résultat de Safarevic

It is shown that if the dilatation tensor G f of a space quasi regular mapping f belongs to the space VMO (vanishing mean oscillation), then f is a local homeomorphism. The same is true If the BMO-norm of G _f is small or if G_f is only close to the VMO space in the BMO-norm.     

Monic divisors with one invariant factor of polynomial matrices

For a nonsingular polynomial matrix over an infinite field, necessary and sufficient conditions for separation of a monic factor from a matrix polynomial with canonical diagonal form diag (1,…,1, φ(x)) are established.     

Zeros of Polynomials on Banach Spaces: The Real Story

Let E be a real Banach space. We show that either E admits a positive definite 2-homogeneous polynomial or every 2-homogeneous polynomial on E has an infinite dimensional subspace on which it is identically zero. Under addition assumptions, we show that such subspaces are non-separable. We examine analogous results for nuclear and absolutely (1,2)-summing 2-homogeneous polynomials and give necessary and sufficient conditions on a compact set K so that C(K) admits a positive definite 2-homogeneous polynomial or a positive definite nuclear 2-homogeneous polynomial.     

Highly Arc-Transitive Digraphs With No Homomorphism Onto <Emphasis Type="Bold">Z</Emphasis>

In an infinite digraph D, an edge e' is reachable from an edge e if there exists an alternating walk in D whose initial and terminal edges are e and e'. Reachability is an equivalence relation and if D is 1-arc-transitive, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs. In Combinatorica, 13 (1993), Cameron, Praeger and Wormald asked if there exist highly arc-transitive digraphs (apart from directed cycles) for which the reachability relation is not universal and which do not have a homomorphism onto the two-way infinite directed path (a Cayley digraph of Z with respect to one generator). In view of an earlier result of Praeger in Australas. J. Combin., 3 (1991), such digraphs are either locally infinite or have equal in- and out-degree. In European J. Combin., 18 (1997), Evans gave an affirmative answer by constructing a locally infinite example. For each odd integer n >= 3, a construction of a highly arc-transitive digraph without property Z satisfying the additional properties that its in- and out-degree are equal to 2 and that the reachability equivalence classes induce alternating cycles of length 2n, is given. Furthermore, using the line digraph operator, digraphs having the above properties but with alternating cycles of length 4 are obtained. Received April 12, 1999 Supported in part by "Ministrstvo za šolstvo, znanost in šport Slovenije", research program PO-0506-0101-99.     

Towards Characterization of All 3 × 3 Extremal Quasiconvex Quadratic Forms

Given a d × d quasiconvex quadratic form, d ≥ 3, we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero, then the form itself is an extremal quasiconvex quadratic form; i.e., it loses its quasiconvexity whenever a convex quadratic form is subtracted from it. In the special case d=3, we slightly weaken the condition; namely we prove that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square, then the form itself is an extremal quadratic form. In the case d=3 we also prove that if the determinant of the acoustic tensor of the form is identically zero, then the form is either extremal or polyconvex. Also, if the determinant of the acoustic tensor of the form is a perfect square, then the form is either extremal or polycovex, or is a sum of a rank‐1 form and an extremal whose acoustic tensor determinant is identically zero. Here we use the notion of extremality introduced by Milton in 1990. © 2017 Wiley Periodicals, Inc.