Related papers: A direct approach to the rational normal form

We present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms. Beginning from basic results developed by Riehm, our solution to this problem hinges on the classification of…

The main goal of this article is to study the existence of a unique positive definite common solution to a pair of matrix equations of the form \begin{eqnarray*} X^r=Q_1 + \displaystyle \sum_{i=1}^{m} {A_i}^*F(X)A_i \mbox{ and } X^s=Q_2 +…

We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…

Arguably the simplest variation of this style of proof as we avoid reducing to the cubic case entirely.

In this paper, we define the \textit{normal form} of collections of disjoint three \textit{bridge arcs} for a given rational $3$-tangle. We show that there is a sequence of \textit{normal jump moves} which leads one to the other for two…

In this paper, we extend the notion of Lyndon word to transfinite words. We prove two main results. We first show that, given a transfinite word, there exists a unique factorization in Lyndon words that are densely non-increasing, a…

An integral quadratic form is called strictly $n$-regular if it primitively represents all quadratic forms in $n$ variables that are primitively represented by its genus. For any $n \geq 2$, it will be shown that there are only finitely…

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. Its existence is equivalent to the existence of a perfect cuboid with all…

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

The concept of unique normal form is formulated in terms of a spectral sequence. As an illustration of this technique some results of Baider and Churchill concerning the normal form of the anharmonic oscillator are reproduced. The aim of…

We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.

A (positive definite and non-classic integral) quadratic form is called strongly $s$-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any…

We survey some results on real rational surfaces focused on their topology and their birational geometry.

We show that a real rational (over $\C$) surfaces are quasi-simple, i.e., that such a surface is determined up to deformation in the class of real surfaces by the topological type of its real structure.

This paper proposes a unique optimization approach for estimating the minimax rational approximation and its application for evaluating matrix functions. Our method enables the extension to generalized rational approximations and has the…

We classify rational, irreducible quartic symmetroids in projective 3-space. They are either singular along a line or a smooth conic section, or they have a triple point or a tacnode.

Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are…

A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.

A rational homogeneous (of degree one) positive real matrix-valued function is presented as the Schur complement of a block of the linear pencil with positive semidefinite matrix coefficients. The partial derivative numerators of a rational…

In this paper, we prove several generalizations and applications of a fixed point theorem. This theorem is used to prove the existence and uniqueness of solutions of the linear sparse matrix problem considered.