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We determine the possible functions that can occur, up to asymptotic equivalence, as growth functions of semigroups, hereditary languages, and algebras. [ABSTRACT FROM AUTHOR]

Let k be a field and let R be a left noetherian k -algebra. The algebra R satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the Spec (R) and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if R and S are two left noetherian k -algebras with dim k (R) , dim k (S) < | k | then if R and S have homeomorphic spectra then R satisfies the Dixmier-Moeglin equivalence if and only if S does. In particular, the topology of Spec (R) can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if k is uncountable and R is affine noetherian and its prime spectrum is a finite disjoint union of locally closed subspaces that are each homeomorphic to the spectrum of an affine commutative ring then R satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if k is countable and R is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that R and S are left noetherian k -algebras that satisfy the Dixmier-Moeglin equivalence then R ⊗ k S does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that eRe also satisfies the Dixmier-Moeglin equivalence, where e is a nonzero idempotent of R. [ABSTRACT FROM AUTHOR]

Let G be a semiabelian variety defined over a field of characteristic 0, endowed with an endomorphism Φ. We prove there is no proper subvariety Y ⊂ G which intersects the orbit of each periodic point of G under the action of Φ. As an application, we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras. [ABSTRACT FROM AUTHOR]

Let k be a field and let A = ⨁ n ≥ 1 A n be a positively graded k -algebra. We recall that A is graded nilpotent if for every d ≥ 1 , the subalgebra of A generated by elements of degree d is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k -subalgebra on two generators. [ABSTRACT FROM AUTHOR]

Leavitt path algebras of an arbitrary graph over a field satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When is a finite graph, satisfying a polynomial identity is shown to be equivalent to the Gelfand-Kirillov dimension of being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph , the Leavitt path algebra has Gelfand-Kirillov dimension zero if and only if has no cycles. Likewise, has Gelfand-Kirillov dimension one if and only if contains at least one cycle, but no cycle in has an exit. [ABSTRACT FROM AUTHOR]

A 3-phase Barker array is a matrix of third roots of unity for which all out-of-phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two-dimensional 3-phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3-phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double-exponentially growing arithmetic function T such that no 3-phase Barker array of size [ABSTRACT FROM AUTHOR]

Abstract: Let be a division ring. We say that is left algebraic over a (not necessarily central) subfield of if every satisfies a polynomial equation with . We show that if is a division ring that is left algebraic over a subfield of bounded degree then is at most -dimensional over its center. This generalizes a result of Kaplansky. For the proof we give a new version of the combinatorial theorem of Shirshov that sufficiently long words over a finite alphabet contain either a -decomposable subword or a high power of a non-trivial subword. We show that if the word does not contain high powers then the factors in the -decomposition may be chosen to be of almost the same length. We conclude by giving a list of problems for algebras that are left algebraic over a commutative subring. [Copyright &y& Elsevier]

Let E be any directed graph, and K be any field. For any ideal I of the Leavitt path algebra LK(E) we provide an explicit description of a set of generators for I. This description allows us to classify the two-sided noetherian Leavitt path algebras over arbitrary graphs. This extends similar results previously known only in the row-finite case. We provide a number of additional consequences of this description, including an identification of those Leavitt path algebras for which all two-sided ideals are graded. Finally, we classify the two-sided artinian Leavitt path algebras over arbitrary graphs. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic p>0 and let f( t,..., t) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K( t,..., t). We prove a generalization of both Derksen's recent analogue of the Skolem-Mahler-Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices ( n,..., n)∈ℕ for which the coefficient of $t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}$ in f( t,..., t) is zero is a p-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to S-unit equations and more generally to the Mordell-Lang Theorem over fields of positive characteristic. [ABSTRACT FROM AUTHOR]

Abstract: Lenagan and Smoktunowicz (2007) (see also Lenagan, Smoktunowicz and Young (in press) ) gave an example of a nil algebra of finite Gelfand–Kirillov dimension. Their construction requires a countable base field, however. We show that for any field k and any monotonically increasing function which grows super-polynomially but subexponentially there exists an infinite-dimensional finitely generated nil k-algebra whose growth is asymptotically bounded by . This construction gives the first examples of nil algebras of subexponential growth over uncountable fields. [Copyright &y& Elsevier]

Abstract: We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show that this result holds for the class of automaton algebras; that is, monomial algebras that have a basis consisting of the set of words recognized by some finite state automaton. This proves a special case of a conjecture of the first author and Agata Smoktunowicz. [Copyright &y& Elsevier]

We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results, e.g., non-holonomicity of the logarithmic sequence is extended to rational functions involving log n. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at an increasing integer sequence with bounded gap lengths is not holonomic. [ABSTRACT FROM AUTHOR]

Abstract: We study just infinite algebras which remain so upon extension of scalars by arbitrary field extensions. Such rings are called stably just infinite. We show that just infinite rings over algebraically closed fields are stably just infinite provided that the ring is either right noetherian (4.2) or countably generated over a large field (6.4). We give examples to show that, over countable fields, a just infinite algebra which is either affine or non-noetherian need not remain just infinite under extension of scalars. We also give a concrete classification of PI stably just infinite rings (5.5) and give two characterizations of non-PI stably just infinite rings in terms of Martindale''s extended center (3.4), (3.5). [Copyright &y& Elsevier]

Abstract: In the vein of Christol, Kamae, Mendès France and Rauzy, we consider the analogue of a problem of Mahler for rational functions in positive characteristic. To solve this question, we prove an extension of Cobham''s theorem for quasi-automatic functions and use the recent generalization of Christol''s theorem obtained by Kedlaya. [Copyright &y& Elsevier]

Abstract: We study prime algebras of quadratic growth. Our first result is that if A is a prime monomial algebra of quadratic growth then A has finitely many prime ideals P such that has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the non-zero prime ideals P such that has GK dimension 2 is non-zero, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has non-zero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra A of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that has GK dimension 1. [Copyright &y& Elsevier]

Abstract: A sequence is said to be k-automatic if the th term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial , we consider the sequence , where is the p-adic valuation. We show that this sequence is p-regular if and only if factors into a product of polynomials, one of which has no roots in , the other which factors into linear polynomials over . This answers a question of Allouche and Shallit. [Copyright &y& Elsevier]

Abstract: Let k be an uncountable algebraically closed field and let A be a countably generated left Noetherian k-algebra. Then we show that is left Noetherian for any field extension K of k. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over k are finitely generated extensions of k. We give examples which show that need not remain left Noetherian if the hypotheses are weakened. [Copyright &y& Elsevier]

Abstract: We show that each polynomial in having only real zeros is the f-polynomial of a multicomplex. It follows that is also the h-polynomial of a Cohen–Macaulay ring and is the g-polynomial of a simplicial polytope. We conjecture that is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zeros of belong to the real interval . We also show that for fixed d the conjecture can fail for at most finitely many polynomials having the required form. [Copyright &y& Elsevier]

Abstract: A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Allouche and Shallit first defined k-regular sequences as a natural generalization of k-automatic sequences. We study the set of values attained by a k-regular sequence and characterize sets with the property that any k-regular sequence taking values in this set is necessarily k-automatic. In particular, we show that an unbounded regular sequence must have infinitely many composite values, answering a question of Allouche and Shallit. [Copyright &y& Elsevier]

By modifying constructions of Beı˘dar and Small we prove that for countably generated prime F-algebras of finite GK dimension there exists an affinization having finite GK dimension. Using this result we show: for any field there exists a prime affine algebra of GK dimension two that is neither primitive nor PI; for any countable field F there exists a prime affine F-algebra of GK dimension three that has non-nil Jacobson radical; for any countable field F there exists an affine primitive F-algebra of GK dimension at most four with center equal to a polynomial ring; for a countable field F there exists a primitive affine Jacobson F-algebra of GK dimension three that does not satisfy the Nullstellensatz. [Copyright &y& Elsevier]

We show, using affinization, that over any field F there exists a primitive, algebraic affine F-algebra that is not locally finite, answering a question of Kaplansky in the affirmative. [Copyright &y& Elsevier]