Related papers: Polynomial Continued Fractions for Algebraic Numbe…
We construct a class of quadratic irrationals having continued fractions of period $n\geq2$ with "small" partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large" partial…
Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the…
Considering an arbitrary pair of distinct and non constant polynomials, $a$ and $b$ in $\mathbb{F}_2[t]$, we build a continued fraction in $\mathbb{F}_2((1/t))$ whose partial quotients are only equal to $a$ or $b$. In a previous work of the…
In this thesis quadratic and cubic algebras, which are extensions of SU(1,1) and SU(2) are studied in detail, with particular attention being given to their construction, their finite and infinite dimensional irreducible representations and…
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.…
We deal with the algebraicity of an iterated Puiseux series in several variables in terms of the properties of its coefficients. Our aim is to generalize to several variables the results from [HM15]. We show that the algebraicity of such a…
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the…
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…
The properties of continued fractions whose partial quotients belong to a quadratic number field K are distinct from those of classical continued fractions. Unlike classical continued fractions, it is currently impossible to identify…
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…
We consider a special class of periodic continued fractions (called alpha-fractions) and discuss the related algebraic and geometric problems. A classical description of the Jacobi variety of a hyperelliptic curve due to Jacobi naturally…
Continued fractions have been long studied due to their strong properties, such as rational approximation. In this extent, their arithmetic over real numbers has represented an intriguing problem throughout the years. In this paper, we…
It is known that the continued fraction expansion of a real number is periodic if and only if the number is a quadratic irrational. In an attempt to generalize this phenomenon to other settings, Jun-Ichi Tamura and Shin-Ichi Yasutomi have…
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…
In this small paper we bring together various open problems on geometric multidimensional continued fractions.
In this paper we prove that there exist infinitely many integers which can be expressed as a sum of four cubes of polynomials with integer coefficients. We give several identities that express the integers 1 and 2 as a sum of four cubes of…
We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a…
The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several…