Related papers: Polynomial Continued Fractions
The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…
Let $f_1,\dots,f_k\in\mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an integer $n<x$ such that the fractional parts $\|f_i(n)\|\ll x^{c/k}$ for all $1\le i\le k$ and for some constant…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
In this paper, as an analogue of the integer case, we study detailedly the period and the rank of the generalized Fibonacci sequence of polynomials over a finite field modulo an arbitrary polynomial. We establish some formulas to compute…
We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions.…
The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic…
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…
We investigate some properties of the higher continued fractions defined recently by Musiker, Ovenhouse, Schiffler, and Zhang. We prove that the maps defining the higher continued fractions are increasing continuous functions on the…
In this paper, we apply results on number systems based on continued fraction expansions to modular arithmetic. We provide two new algorithms in order to compute modular multiplication and modular division. The presented algorithms are…
A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…
There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…
In this paper we establish properties of independence for the continued fraction expansions of two algebraic numbers. Roughly speaking, if the continued fraction expansions of two irrational algebraic numbers have the same long sub-word,…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…
We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…
In this paper we examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of values of numerators of differences of consecutive elements in this…
We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}.…
We obtain two new algorithms for partial fraction decompositions; the first is over algebraically closed fields, and the second is over general fields. These algorithms takes $O(M^2)$ time, where $M$ is the degree of the denominator of the…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…
We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call `Jones rational coincidences'. We provide moves on the continued fraction expansion of the associated rational which we prove do not…