Related papers: A note on Cardano's formula
Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential…
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.
This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this…
Given any complex number $a$, we prove that there are infinitely many simple roots of the equation $\zeta(s)=a$ with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the…
We obtain exact, simple and very compact expressions for the linearization coefficients of the products of orthogonal polynomials; both the conventional Clebsch-Gordan-type and the modified version. The expressions are general depending…
If the coefficients of polynomials are selected by some random process, the zeros of the resulting polynomials are in some sense random. In this paper the author rephrases the above in more precise language, and calculates the joint…
The papers shows an algorithm to search for approximations of reals to rationals of the form a/b^2 that runs on \sqrt(b) polynomial time steps.
Integral Cauchy theorem is used to derive closed-form expressions of the roots of a univariate polynomial of any degree as integrals of elementary functions.
We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.
We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.
The solution of equations from the title is well known since the Euler's time. However, its proof in the case of multiple roots of the characteristic polynomial is rather long and technical and even appearance of the factors $x^m$ looks…
The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of…
We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…
In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in…
Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…
We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.
A problem concerning the shift of roots of a system of homogeneous algebraic equations is investigated. Its conservation and decomposition of a multiple root into simple roots are discussed.
We show that every monic polynomial of degree three with complex coefficients and no repeated roots is either a (vertical and horizontal) translation of $y=x^3$ or can be composed with a linear function to obtain a Ramanujan cubic. As a…
We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally…
A formulation of Dirac's equation using complex-quaternionic coordinates appears to yield an enormous gain in formal elegance, as there is no longer any need to invoke Dirac matrices. This formulation, however, entails several…