Related papers: A quick route to unique factorization in quadratic…
In this paper, we consider the existence of a factorization of a monic, bounded motion polynomial. We prove existence of factorizations, possibly after multiplication with a real polynomial and provide algorithms for computing polynomial…
Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper…
This note presents the basic mathematical structure of a new integer factorization method based on systems of linear Diophantine equations.
In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.
We discuss the structure of infrared singularities in on-shell QCD amplitudes with massive partons and present a general factorization formula in the limit of small parton masses. The factorization formula gives rise to an all-order…
We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include…
We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…
If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…
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…
We present a simple proof of the factorization of (complex) symmetric matrices into a product of a square matrix and its transpose, and discuss its application in establishing a uniqueness property of certain antilinear operators.
Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a…
In this article we show how the Dedekind-Hasse criterion may be applied to prove a simple result about quadratic number fields that usually is derived as a consequence of the theory of ideals and ideal classes.
Given an ordered structure, we study a natural way to extend the order to preorders on type spaces. For definably complete, linearly ordered structures, we give a characterisation of the preorder on the space of 1-types. We apply these…
We prove a generalization of Orlov's theorem for matrix factorizations with $n$ steps. Let $X$ be a regular scheme, $W\colon X\to \mathbb{A}^1$ a flat morphism and $D:=W^{-1}(0)$ its central fiber. We construct an appropriate triangulated…
We investigate the uniqueness of factorisation of possibly disconnected finite graphs with respect to the Cartesian, the strong and the direct product. It is proved that if a graph has $n$ connected components, where $n$ is prime, or…
We briefly review the calculational procedure for the PQCD prediction for hard exclusive quantities and reconsider the problem of the factorization scale dependence.
In this paper, we derive the quadratic formula as a consequence of constructively proving the existence of standard and factored forms for general form real quadratic functions. Emphasis is put on connections to graphing of corresponding…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…
We give an alternative argument to the application of the so-called Maurey- Nikishin-Pisier factorisation in Fourier restriction theory. Based on an induction-on-scales argument, our comparably simple method applies to any compact quadratic…
Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four. The operators are assumed to have a completely factorable symbol. It is proved that ``irreducible'' parametric…