Search for: All records

Abstract Let $$K$$ be any field, and let $$n$$ be a positive integer. If we denote by $$\xi _{\textrm{SL}_n}\colon \textrm{SL}_n\times \textrm{SL}_n\to \textrm{SL}_n$$ the commutator morphism over $$K$$, then $$\xi _{\textrm{SL}_n}$$ is flat over the complement of the center of $$\textrm{SL}_n$$. 

We define the notion of an almost polynomial identity of an associative algebra R R , and show that its existence implies the existence of an actual polynomial identity of R R . A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras. 

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon, 

This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry. 

We say a power series ∑ n = 0 ∞ a n q n \sum _{n=0}^\infty a_n q^n is multiplicative if the sequence 1 , a 2 / a 1 , … , a n / a 1 , … 1,a_2/a_1,\ldots ,a_n/a_1,\ldots is so. In this paper, we consider multiplicative power series f f such that f 2 f^2 is also multiplicative. We find a number of examples for which f f is a rational function or a theta series and prove that the complete set of solutions is the locus of a (probably reducible) affine variety over C \mathbb {C} . The precise determination of this variety turns out to be a finite computational problem, but it seems to be beyond the reach of current computer algebra systems. The proof of the theorem depends on a bound on the logarithmic capacity of the Mandelbrot set. 

Abstract For $$G=\textrm{GL}(n,q)$$, the proportion $$P_{n,q}$$ of pairs $$(\chi ,g)$$ in $$\textrm{Irr}(G)\times G$$ with $$\chi (g)\neq 0$$ satisfies $$P_{n,q}\to 0$$ as $$n\to \infty $$. 

Let f∈ℚ(x) be a non-constant rational function. We consider ‘Waring’s problem for f(x), i.e., whether every element of ℚ can be written as a bounded sum of elements of {f(a)∣a∈ℚ}. For rational functions of degree 2, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring’s problem. We also consider the 'easier Waring’s problem': whether every element of ℚ can be represented as a bounded sum of elements of {±f(a)∣a∈ℚ}. ⁠.