A model for the relatively free graded algebra of block triangular matrices with entries from a graded algebra

Let G be a group and A be a G-graded algebra satisfying a polynomial identity. We buid up a model for the relative free G-graded algebra and we obtain, as an application, the "factoring" property for the T_G-ideals of block triangular matrices with entries from the finite dimensional Grassmann algebra E for some particular Z_2-grading

Quantum measurements and the Abelian Stabilizer Problem

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.Comment: 22 pages, LATE

On a family of linear recurrences

We concern ourselves with the family of linear recurrence relations aj = aj−1+aj−d with the initial conditions a0 = . . . = ad−1 = 1. We discuss the periodicity evaluation of such recurrences for prime powers d, and demonstrate that a key feature of our evaluation method relates to an instance of Shor's algorithm for factoring. As an application, we discuss how efficient quantum circuit designs may be completely recast as a problem relating to linear recurrence relations

Maps between curves and arithmetic obstructions

Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.Comment: 8 page