Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector of coefficients of the polynomial as a word on a ternary alphabet ${-1,0 ,+1}$. It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.
We show that, in an alphabet of $n$ symbols, the number of words of length $n$ whose number of different symbols is away from $(1-1/e)n$, which is the value expected by the Poisson distribution, has exponential decay in $n$. We use Laplaces method for sums and known bounds of Stirling numbers of the second kind. We express our result in terms of inequalities.
We study the probabilistic behaviour of the continued fraction expansion of a quadratic irrational number, when weighted by some additive cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with the underlying dynamical system associated with the Gauss map, and its weighted periodic trajectories. We work with analytic combinatorics methods, and mainly with bivariate Dirichlet generating functions; we use various tools, from number theory (the Landau Theorem), from probability (the Quasi-Powers Theorem), or from dynamical systems: our main object of study is the (weighted) transfer operator, that we relate with the generating functions of interest. The present paper exhibits a strong parallelism with the methods which have been previously introduced by Baladi and Vallee in the study of rational trajectories. However, the present study is more involved and uses a deeper functional analysis framework.
We obtain estimates on the number $|mathcal{A}_{boldsymbol{lambda}}|$ of elements on a linear family $mathcal{A}$ of monic polynomials of $mathbb{F}_q[T]$ of degree $n$ having factorization pattern $boldsymbol{lambda}:=1^{lambda_1}2^{lambda_2}cdots n^{lambda_n}$. We show that $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}/{2}})$, where $mathcal{T}(boldsymbol{lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $boldsymbol{lambda}$ and $m$ is the codimension of $mathcal{A}$. Furthermore, if the family $mathcal{A}$ under consideration is sparse, then $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $mathcal{O}$--notation in terms of $boldsymbol{lambda}$ and $mathcal{A}$ with good behavior. Our approach reduces the question to estimate the number of $mathbb{F}_q$--rational points of certain families of complete intersections defined over $mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $mathbb{F}_q$--rational points are established.
Most hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible.
We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},..., a_{d-s} are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d,s,bfs{a})=mu_d.q+mathcal{O}(1), where V(d,s,bfs{a}) is such an average cardinality, mu_d:=sum_{r=1}^d{(-1)^{r-1}}/{r!} and bfs{a}:=(a_{d-1},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the mathcal{O}--notation in terms of d and s with good behavior. Our approach reduces the question to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq--rational points is established.
This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$. Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$. By imposing $n_{_N}le m forall Nin zN$, Jarnik defined the corresponding sets $E_m$ and gave a first estimate of $d_H(E_m)$, $d_H$ the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with $d_H(E_m)$ and $d_H(F_m)$, $F_m$ being the set of real numbers for which ${sum_{i=1}^N n_iover N}le m$.
