Related papers: Growth for the central polynomials

Let $f$ and $g$ be two noncommutative polynomials in disjoint sets of variables. An algebra $A$ is verbally prime if whenever $f\cdot g$ is an identity for $A$ then either $f$ or $g$ is also an identity. As an analogue of this property…

In this paper, we study the growth of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)\Delta^mf(z)+\cdots+P_1(z)\Delta f(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for…

In 1988, S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M_2(k)) is finitely based, and he raised the question as to whether CP(A) is finitely based for every (unitary) associative algebra A with nonzero…

Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean of the…

In recent years, the study of the $T$-space of central polynomials of an algebra $A$ has become an object of great interest in the PI-theory. Such interest has been extended to the context of algebras with additional structures. The main…

We study the dynamics of polynomial mappings f:C^k to C^k of degree at least 2 that extend continuously to projective space P^k. Our approach is to study the dynamics near the hyperplane at infinity and then making a descent to K --- the…

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely generated graded $S$-module, the Hilbert coefficients $e_i(M/I^kM)$ are polynomial…

Let $\mathbb{K}$ be a field of characteristic zero and $B=B_0+B_1$ a finite dimensional associative superalgebra. In this paper we investigate the polynomial identities of the relatively free algebras of finite rank of the variety…

Let $A$ be an associative algebra graded by a finite group $G$ over a field ${F}$ of characteristic zero. One associates to $A$ the sequence of $G$-graded codimensions $c_n^G(A)$, $n=1,2,\ldots$, which measures the growth of the polynomial…

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in…

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any…

Consider the interval of integers $I_{m,n} = \{m, m+1, m+2,\ldots, m+n-1 \}$. For fixed integers $h,k,m$, and $c$, let $\Phi_{h,k,m}^{(c)}(n)$ denote the number of solutions of the equation $(a_1+\cdots + a_h)- (a_{h+1} + \cdots +…

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 $A/P$ has GK dimension one. This shows that prime monomial…

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive…

Let $F$ be a finite field with characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis of the $\mathbb{Z}_{2}$-graded polynomial…

Assume that $X= {x_1,...,x_g}$ is a finite alphabet and $K$ is a field. We study monomial algebras $A= K <X> /(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincar\'{e}-Birkhoff-Witt type basis…

Let f and g be nonconstant polynomials over an arbitrary field K. In this paper we study the intersection of the polynomial rings K[f] and K[g], and in particular we ask whether this intersection is larger than K. We completely resolve this…

Let $F$ be an infinite field of characteristic different from 2, and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the $\mathbb{Z}$-graded polynomial identities of $E$ with respect…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials…