Let q be a power of an odd prime and let k, n ∈ N such that 1 < k ≤ n. We investigate the existence of self-reciprocal irreducible monic polynomials over Fq, of degree 2n and their k-th coefficient prescribed.

Let G be a group and G′ its commutator subgroup. Denote by c G the minimal number such that every element ofG′ can be expressed as a product of at most c G commutators. A group G is called a c-group if c G is finite. For any positive integer n, denote by cn the class of groups with commutator length, c G n. Let Fn,t 〈x1, . . . , xn〉 andMn,t 〈x1, . . . , xn〉 be, respectively, the free nilpotent ...

This paper presents a method for designing a ®xed controller, which is able to control in a stable fashion a family of linear time-invariant n-th order plants with arbitrary relative degree. This plant family is de®ned in terms of a nominal transfer function of rational type and bounded variations of the coef®cients of numerator and denominator polynomials. The design method is based on the alg...

Let be a d-dimensional simplex with vertices v 0 ; ; v d and B n (f;) denote the n th degree Bernstein polynomial of a continuous function f on. Dahmen and Micchelli 4] proved that B n (f;) B n+1 (f;); n 2 N; for any convex function f on , and it is clear that a necessary and suucient condition for the inequality to become an identity for all n 2 N is that f is an aane polynomial. Let m be the ...

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space Pn−1. In this paper, we achieve Serre’s conjecture in the special case of smooth cyclic covers of any degree when n ≥ 10, and surpass it for covers of degree r ≥ 3 when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsin...

By the Shepherd–Leedham-Green–McKay theorem on finite p-groups of maximal class, if a finite p-group of order pn has nilpotency class n−1, then it has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Counterexamples to a rank analogue of this theorem are constructed, which give a negative solution to Problem 16.103 in Kourovka Notebook. Moreover, it is shown that there...

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies given polynomial identity r(t) = 0. In the special case of r=tn−1 we obtain uniform bound on class periodic derivation order n. even find optimal in characteristic p if does not divide certain invariant ρn.