Narrow Results

The additive primitive length of an element $f$ of a relatively free algebra $F_d(𝔙)$ in a variety of algebras $𝔙$ is equal to the minimal number $\ell$ such that $f$ can be presented as a sum of $\ell$ primitive elements. We give an upper bound for the additive primitive length of the elements in the $d$-generated polynomial algebra over a field of characteristic 0, $d>1$. The bound depends on $d$ and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free $d$-generated nilpotent-by-abelian Lie algebras is bounded by 5 for $d=3$ and by 6 for $d>3$. If the field has two elements only, then our bounds are 6 for $d=3$ and 7 for $d>3$. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.

Motivated by results about “untangling” closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, $d_{\mathrm{\text{prim}}}(g;F_N)$ and $d_{\mathrm{\text{simp}}}(g;F_N)$, where $1\ne g\in F_N$, and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence $d_{\mathrm{\text{prim}}}(a^n{b}^n;F(a,b))$ as $n\to \infty$. Answering a question from [17], we prove that this sequence is unbounded and that for $n_i=\mathrm{\text{lcm}}(1,2,\dots ,i)$, we have $|d_{\mathrm{\text{prim}}}(a^{n_i}{b}^{n_i};F(a,b))-log(n_i)|=o(log(n_i))$. By contrast, we show that for all $n\ge 2$, one has $d_{\mathrm{\text{simp}}}(a^n{b}^n;F(a,b))=2$. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.

D. Puder defined the primitivity rank of elements of free groups [Primitive words, free factors and measure preservation, Israel J. Math.201(1) (2014) 25–73], we give a similar definition for free algebras of Schreier varieties and prove properties of a primitivity rank using the properties of the almost primitive elements.

Let $F_n$ be a free Lie algebra of finite rank $n,$$n\ge 2$. We give another proof of the following criterion which is proven by Mikhalev and Zolotykh, using the idea of $k$-primitivity: An endomorphism of $F_n$ preserving primitivity of elements is an automorphism.

In this paper, we consider rational functions $f$ with some minor restrictions over the finite field ${𝔽}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a pair $(\alpha ,f(\alpha ))$ of primitive normal elements in ${𝔽}_{q^n}$ over ${𝔽}_q.$ Moreover, for $q=2^k$ and rational functions $f$ with quadratic numerators and denominators, we explicitly find that there are at most $55$ finite fields ${𝔽}_{q^n}$ in which such a pair $(\alpha ,f(\alpha ))$ of primitive normal elements may not exist.

This paper provides a mean value theorem for arithmetic functions $f$ defined by

In this paper, we study several topics on additive decompositions of primitive elements in finite fields. Also we refine some bounds obtained by Dartyge and Sárközy as well as Shparlinski.

Let ${\mathrm{\text{BPU}}}_n$ be the classifying space of ${\mathrm{\text{PU}}}_n$, the projective unitary group of order $n$, for $n>1$. We use a Serre spectral sequence to determine the ring structure of $H^{\ast }({\mathrm{\text{BPU}}}_n;\mathbb{Z})$ up to degree $10$, as well as a family of distinguished elements of $H^{2p+2}({\mathrm{\text{BPU}}}_n;\mathbb{Z})$, for each prime divisor $p$ of $n$. We also study the primitive elements of $H^{\ast }({\mathrm{\text{BU}}}_n;\mathbb{Z})$ as a comodule over $H^{\ast }(K(\mathbb{Z},2);\mathbb{Z})$, where the comodule structure is given by an action of $K(\mathbb{Z},2)\simeq {\mathrm{\text{BS}}}^1$ on ${\mathrm{\text{BU}}}_n$ corresponding to the action of taking the tensor product of a complex line bundle and an $n$-dimensional complex vector bundle.