On the Linear Independence of Roots

Linearly Unrelated Sequences

There are not many new results concerning the linear independence of numbers. Exceptions in the last decade are, e.g., the result of Sorokin [8] which proves the linear independence of logarithmus of special rational numbers, or that of Bezivin [2] which proves linear independence of roots of special functional equations. The algebraic independence of numbers can be considered as a generalizati...

On the Roots of Hosoya Polynomial of a Graph

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x , where, d(u ,v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific g...

On a Sufficient and Necessary Condition for a Multivariate Polynomial to Have Algebraically Dependent Roots - an Elementary Proof

In the paper we prove that a multivariate polynomial has algebraically dependent roots iff the coefficients are algebraic numbers up to a common proportional term. A complex analytic proof can be found in [2] with applications in the theory of linear functional equations, see also [3, an open problem, section 4.4] and [1]. Here we present an elementary proof involving cardinality properties and...

On group Elements Having Square Roots

Application of the linear Differential Equations on the Plane and Elements of Nonlinear Systems, In Economics

In recent years, it has become increasingly important to incorporate explicit dynamics in economic analysis. These two tools that mathematicians have developed, differential equations and optimal control theory, are probably the most basic for economists to analyze dynamic problems. In this paper I will consider the linear differential equations on the plane (phase diagram) and elements of nonl...