Related papers: On the Complete Solution to the Most General Fifth…
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a…
In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in several variables. This algorithm is a generalization of a method developed for…
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…
In the first part of the paper, we address an invertible matrix polynomial $L(z)$ and its inverse $\hat{L}(z) := -L(z)^{-1}$. We present a method for obtaining a canonical set of root functions and Jordan chains of $L(z)$ through elementary…
We present a simple method to decompose the Green forms corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are…
An effective method for generating linear equations of maximal symmetry in their much general normal form is obtained. In the said normal form, the coefficients of the equation are differential functions of the coefficient of the term of…
The roots of any polynomial of degree m with complex integer coefficients can be computed by manipulation of sequences made from distinct symbols and counting the different symbols in the sequences. This method requires only primitive…
The large sieve is used to estimate the density of integral quadratic polynomials $Q$, such that there exists an odd degree integral polynomial which has resultant $\pm 1$ with $Q$. Given a monic integral polynomial $R$ of odd degree, this…
We raise a question on the existence of continuous roots of families of monic polynomials (by the root of a family of polynomials we mean a function of the coefficients of polynomials of a given family that maps each tuple of coefficients…
We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the…
Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of…
In this paper, an efficient method is presented for solving three dimensional Volterra integral equations of the second kind with continuous kernel. Shifted Chebyshev polynomial is applied to approximate a solution for these integral…
Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a…
We consider a certain mixed polynomial which is an extended Lens equation $L_{n,m}=\bar z^m-p(z)/q(z)$ with $\text{degree}\, q=n$, $\text{degree}\, p<n$ whose numerator is a mixed polynomial of degree $(n+m;n,m)$. Then we consider its…
An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas…
This short article presents explicit expressions for roots of a quartic equation that has all four real roots. Although a general expression for quartic roots is available on Wikipedia, an optimized and slightly shorter expression for only…
The solution of the cubic equation has a century-long history; however, the usual presentation is geared towards applications in algebra and is somewhat inconvenient to use in optimization where frequently the main interest lies in real…
We consider the even monic degree-$10$ second cuboid polynomial $Q_{p,q}(t)\in\mathbb{Z}[t]$ depending on coprime integers $p\neq q>0$. We exclude the existence of a splitting of type $5+5$ over $\mathbb{Q}$, i.e., a factorization of…
We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Pl\"ucker formulas are polynomials in the degree, just as the…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…