Related papers: The quadratic formula made hard: A less radical ap…
A method to the explict solutions of general systems of algebraic equations is presented via the metric form of affiliated K\"ahler manifolds. The solutions to these systems arise from sets of geodesic second order non-linear differential…
We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center…
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…
This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type $_2F_{1}(z)$, for certain…
Classical Calogero-Moser models with rational potential are known to be superintegrable. That is, on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom, they possess an…
We propose a new approach at Fermat's Last Theorem (FLT) solution: for each FLT equation we associate a polynomial of the same degree. The study of the roots of the polynomial allows us to investigate the FLT validity. This technique,…
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and…
The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…
The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…
The aim of this paper consists of providing summation formulas for the $k$-Fibonacci numbers ($k \in \mathbb{Z}$, $k \geq 2$) and their asymptotic equivalents in terms of generalized binomial coefficients. Our main tools are the Lagrange…
We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a standalone method of computation of Gaussian…
We study a q-generalization of the classical Laguerre/Hermite orthogonal polynomials. Explicit results include: the recursive coefficients, matrix elements of generators for the Heisenberg algebra, and the Hankel determinants. The power of…
Given a subset of $\mathbb C$ containing $x,y$, one can add $x + y,\,x - y,\,xy$ or (when $y\ne0$) $x/y$ or any $z$ such that $z^2=x$. Let $p$ be a prime Fermat number. We prove that it is possible to obtain from $\{1\}$ a set containing…
We give a modern approach to the famous Cardano and Ferrari formulas in the algebraic equations with three and four degrees. Namely, we reconstruct these formulas from the point of view of superposition principle in quantum computation…
We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…
This paper applies the modular approach to obtain effectively computable bounds for Fermat-type equations over number fields, while also discussing the differences and obstructions that arise when considering such equations over totally…
The radical solution of polynomials with rational coefficients is a famous solved problem. This paper found that it is a $\mathbb{NP}$ problem. Furthermore, this paper found that arbitrary $ \mathscr{P} \in \mathbb{P}$ shall have a one-way…