arXiv subjects
Publications and source records attributed to Jason Bell.
We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 25 is the sum of at most three natural numbers whose base-2 representation has an equal number of 0's and 1's.
We characterize those $k$-automatic sets $S$ of natural numbers that form an additive basis for the natural numbers, and we show that this characterization is effective. In addition, we give an algorithm to determine the smallest $j$ such that $S$ forms an additive basis of order $j$, if it exists.
A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for $D$-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if $R$ is a commutative affine Hopf algebra over a field of characteristic zero, and $A$ is an Ore extension to which the Hopf algebra structure extends, then $A$ satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such $A$ are Hopf Ore extensions
A noncommutative analogue of the Zariski cancellation problem asks whether $A[x]\cong B[x]$ implies $A\cong B$ when $A$ and $B$ are noncommutative algebras. We resolve this affirmatively in the case when $A$ is a noncommutative finitely generated domain over the complex field of Gelfand-Kirillov dimension two. In addition, we resolve the Zariski cancellation problem for several classes of Artin-Schelter regular algebras of higher Gelfand-Kirillov dimension.
Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to each of these three properties, then we say that the algebra satisfies the strong Dixmier-Moeglin equivalence. Using the example of the universal enveloping algebra of sl_2(C), we show that the strong Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie algebra g, a non root of unity q\neq 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subalgebra U_q[w] of the quantised enveloping K-algebra U_q(g). These quantum Schubert cells U_q[w] are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the strong Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier-Moeglin equivalence.
We study how well Fekete polynomials $$ F_p(X) = \sum_{n=0}^{p-1} \left(\frac{n}{p}\right) X^n \in {\mathbb Z}[X] $$ with the coefficients given by Legendre symbols modulo a prime $p$, can be approximated by power series representing algebraic functions of a given degree. We also obtain some explicit results describing polynomial recurrence relations which are satisfied by the coefficients of such algebraic functions.
An algebra $A$ satisfies the Dixmier-Moeglin equivalence if we have the equivalences: $$P~{\rm primitive}\iff P~{\rm rational}\iff P ~{\rm locally~closed~}\qquad~{\rm for}~P\in {\rm Spec}(A).$$ We study the robustness of the Dixmier-Moeglin equivalence under extension of scalars and under the formation of Ore extensions. In particular, we show that the Dixmier-Moeglin equivalence is preserved under base change for finitely generated complex noetherian algebras. We also study Ore extensions of finitely generated complex noetherian algebras $A$. If $T:A\to A$ is either a $\mathbb{C}$-algebra automorphism or a $\mathbb{C}$-linear derivation of $A$, we say that $T$ is \emph{frame-preserving} if there exists a finite-dimensional subspace $V\subseteq A$ that generates $A$ as an algebra such that $T(V)\subseteq V$. We show that if $A$ is of finite Gelfand-Kirillov dimension and has the property that all prime ideals of $A$ are completely prime and $A$ satisfies the Dixmier-Moeglin equivalence then the Ore extension $A[x;T]$ satisfies the Dixmier-Moeglin equivalence whenever $T$ is a frame-preserving derivation or automorphism.
Let $A$ and $B$ be two connected graded algebras finitely generated in degree one. If $A$ is isomorphic to $B$ as ungraded algebras, then they are also isomorphic to each other as graded algebras.
Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite $\operatorname{GK}$ dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.
Suppose that $F(x)\in\mathbb{Z}[[x]]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$. In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If $F(x)$ is also regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$-number or a $T$-number.
Leavitt path algebras L of an arbitrary graph E over a field K satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When E is a finite graph, L satisfying a polynomial identity is shown to be equivalent to the Gelfand-Kirillov dimension of L being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph E, the Leavitt path algebra L has Gelfand-Kirillov dimension zero if and only if E has no cycles. Likewise, L has Gelfand-Kirillov dimension one if and only if E contains at least one cycle, but no cycle in E has an exit.
The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if and only either (a) m is not equal to n or (b) m=n and the bases (ordered pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory.
We present results and background rationale in support of a Polya--Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of dynamical zeta functions of compact group automorphisms.
Characteristic points have been a primary tool in the study of a generating function defined by a single recursive equation. We investigate the proper way to adapt this tool when working with multi-equation recursive systems.
The aim of this article is to give explicit formulae for various generating functions, including the generating function of torus-invariant primitive ideals in the big cell of the quantum minuscule grassmannian of type B_n.
Let $E$ be any directed graph, and $K$ any field. We classify those graphs $E$ for which the Leavitt path algebra $L_K(E)$ is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.
We present a combinatorial method to determine the dimension of $\C{H}$-strata in the algebra of $m\times n$ quantum matrices $\Oq$ as follows. To a given $\C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the dimension of the $\C{H}$-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the $d$-dimensional $\C{H}$-strata in $\Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors \cite{bldim,bll} regarding the asymptotic proportion of $d$-dimensional $\C{H}$-strata in $\Oq$.
The aim of this paper is to study the representation theory of quantum Schubert cells. Let $\g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $\g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(\g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits a stratification indexed by those primes that are invariant under a natural torus action. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a given torus-invariant prime.