Related papers: On the Linear Independence of Roots

We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory…

A skew polynomial ring $R=K[x;\sigma,\delta]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A…

The purpose of this paper is to study fields whose multiplicative groups admit the structure of linear spaces. We prove that the multiplicative group of a finite field is a linear space if and only if the order of the multiplicative group…

Positive bases, which play a key role in understanding derivative free optimization methods that use a direct search framework, are positive spanning sets that are positively linearly independent. The cardinality of a positive basis in…

We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.

It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple…

Let $q$ be a Pisot or Salem number. Let $f_j(x)$ $(j=1,2,\dots)$ be integer-valued polynomials of degree $\ge2$ with positive leading coefficients, and let $\{a_j (n)\}_{n\ge1}$ $(j=1,2,\dots)$ be sequences of algebraic integers in the…

We prove that all arrangements (consistent with the Rolle theorem and some other natural restrictions) of the real roots of a real polynomial and of its $s$-th derivative are realizable by real polynomials.

The roots of a complex polynomial depend continuously on the coefficients; that is, an infinitesimal perturbation of the coefficients results in an infinitesimal perturbation of the roots. A short, straightforward proof of this is possible…

In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.

A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…

Let $(i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}$ and $S_{i,j}$ be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over $\mathbb{Q}$ of the…

This survey revolves around the question how the roots of a monic polynomial (resp. the spectral decomposition of a linear operator), whose coefficients depend in a smooth way on parameters, depend on those parameters. The parameter…

We explore when it is legal to differentiate a polynomial evaluated at a root of unity using modular arithmetic.

This paper provides the first criteria for the linear independence of multiple polylogarithm values over algebraic number fields. In particular, we derive novel results regarding the linear independence of products of polylogarithms at…

We extend the theory of d-separation to cases in which data instances are not independent and identically distributed. We show that applying the rules of d-separation directly to the structure of probabilistic models of relational data…

We introduce the concepts of dependence and independence in a very general framework. We use a concept of rank to study dependence and independence. By means of the rank we identify (total) dependence with inability to create more…

Let $H_n =\sum\limits_{k=1}^n \frac{1}{k}$ be the $n$-th harmonic number. Euler extended it to complex arguments and defined $H_r$ for any complex number $r$ except for the negative integers. In this paper, we give a new proof of the…

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…

Deciding whether or not two polynomials have isomoprhic splitting fields over the rationals is the Field Isomorphism Problem. We consider polynomials of the form $f_n(x) = x^4-nx^3-6x^2+nx+1$ with $n \neq 3$ a positive integer and we let…