Related papers: Solving cubic equations by completing the cube and…
One can hardly believe that there is still something to be said about cubic equations. To dodge this doubt, we will instead try and say something about Sylvester. He doubtless found a way of solving cubic equations. As mentioned by Rota, it…
In this paper we show that Cardanos formula for the solution of cubic equations can be reduced to expressions involving only square roots if the real root is rational.
Motivated by the recent work of William Y.C. Chen, in which he presents a way to solve cubic equations by considering the identity of Sylvester, we investigate the solutions obtained in this way. It leads us to a unified expression of the…
Building on a classification of zeros of cubic equations due to the $12$-th century Persian mathematician Sharaf al-Din Tusi, together with Smale's theory of {\it point estimation}, we derive an efficient recipe for computing high-precision…
One of the main themes in this thesis is the description of the signature of both the infinite place and the finite places in cubic function fields of any characteristic and quartic function fields of characteristic at least 5. For these…
This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the…
We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this…
Let $p(z)$ be a monic cubic complex polynomial with distinct roots and distinct critical points. We say a critical point has the {\it Voronoi property} if it lies in the Voronoi cell of a root $\theta$, $V(\theta)$, i.e. the set of points…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
The paper deals with the resolution of third and fourth degree equations by means of radicals. It is a survey of some historical details about this fundamental problem. Moreover, it explains practical methods for the resolution of third and…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four quadratic equations with respect to six…
We develop an operator algebraic framework for generalized Cardano polynomials and show how their structure naturally leads to an operator formulation of Cardano method that is compatible with tools and concepts from quantum information…
An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this…
We study the equation $(x-4r)^3 + (x-3r)^3 + (x-2r)^3+(x-r)^3 + x^3 + (x+r)^3+(x+2r)^3 + (x+3r)^3 + (x+4r)^3 = y^p$, which is a natural continuation of previous works carried out by A. Arg\'{a}ez-Garc\'{i}a and the fourth author (perfect…
We consider a sequence of sums of powers of the the roots of the cubic equation characterizing the Tribonacci sequences and derive its relationship with a particular Tribonacci sequence. Then we make a conjecture on the possible…
We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…
This article introduces an intuitive function MY that simplifies solving cubic equations without venturing into the complex space. To many, it's quite strange that cubic root(s) are expressed using trigonometric functions in the…
A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in…
The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there…
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered…