It is shown that only a fraction of 2 −Ω(n) of the graphs on n vertices have an integral spectrum. Although there are several explicit constructions of such graphs, no upper bound for their number has been known. Graphs of this type play an important role in quantum networks supporting the so-called perfect state transfer.
For a prime p, we consider some natural classes of matrices over a finite field F p of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of prescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with components in a given subinterval [−H, H] ⊆ [−(p − 1)/2, (p − 1)/2] is asymptotically close to the expected value.
We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.Nous prouvons une borne inférieure pour l'ordre multiplicatif des périodes de Gauss générant les bases normales sur les corps finis. Cette borne améliore une borne antérieure duà J. von zur Gathen et I. E. Shparlinski.
Abstract. We recall that a polynomial f (X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f (X) ∈ [ޚX] are stable over .ޑ We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f (X) ∈ [ޚX] can be detected by a finite algorithm; this property is closely related to the stability of f (X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.
Let C be a smooth absolutely irreducible curve of genus g ≥ 1 defined over F q , the finite field of q elements. Let #C(F q n ) be the number of F q n -rational points on C. Under a certain multiplicative independence condition on the roots of the zetafunction of C, we derive an asymptotic formula for the number of n = 1, . . . , N such that (#C(F q n ) − q n − 1)/2gq n/2 belongs to a given interval I ⊆ [−1, 1]. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve E is defined over Q and considered modulo consecutive primes p, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus g = 2.
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