Related papers: A Visualizable, Constructive Proof of the Fundamen…

We present a short elementary proof of the well-known criterion for a cubic polynomial to have three real roots. The proof is based on Fermat's approach to calculus for polynomials. This approach illustrates the idea of a derivative…

Starting from the solution to Bring's equation the root ambiguity is removed from the solution to the quintic equation. This gives the five complex roots of the quintic equation as indicated by Gauss's Fundamental Theorem of Algebra.r

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of…

The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. The nature of this transcendence is measured by a Galois group built from the Picard-Vessiot theory of…

For any type of fundamental groupoid scheme, we construct an algebraic cohomology theory for varieties with coefficients in the base field. This is a minor variant of \'etale cohomology, involving neither de Rham complexes nor…

We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots.…

In this paper we construct a visualization of the Abel's Impossibility Theorem also known as the Abel-Ruffini Theorem. Using the canvas object in JavaScript along with the p5.js library, and given any expression that uses analytic functions…

Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…

A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…

We present an innovative approach to dimensional analysis, based on a general representation theorem for complete quantity functions admitting a covariant scalar representation; this theorem is in turn grounded in a purely algebraic theory…

This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…

Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.

This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial. They can be approximated at a low computational cost if the…

This paper presents a path to proving the Four-Color Theorem that differs from the traditional "reducible configuration" method. By introducing concepts such as "outer boundary," "primitive set," "Property A," "knot," "valid pair group,"…

In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev method is developed. The method allows to find simultaneously all roots of the algebraic equation in the case when the roots are supposed to be multiple with…

This paper describes an algorithm for determining the branching geometry of algebraic functions. The graphs of these complex-valued functions have a complicated interweaving structure that can be described by analytic branches separated by…

Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…

An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…

Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but…