On real Waring decompositions of real binary forms

The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial $p$ of degree $d$ as a finite sum of $d$-{th} powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any given real binary form $p$ of length at most its degree. In fact, we construct a semialgebraic family of Waring decompositions for $p$. Some examples are shown to highlight the difference between the real and the complex case.Comment: 21 pages; typos correcte

Problema de Waring para formas binarias reales

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 23-03-2023El Problema de Waring sobre anillos de polinomios aborda el problema de la reescritura de un polinomio homogéneo de grado d como una suma finita de potencias d-ésimas de formas lineales. El objetivo de este trabajo es el estudio de este problema en el caso de polinomios homogéneos reales en dos variables o formas binarias reales. Uno de los aspectos relacionados con este Problema de Waring más detalladamente estudiado consiste en determinar la longitud de estas descomposiciones o bien acotarla. Nuestro trabajo proporciona un método constructivo para obtener una descomposición de Waring real (WD) para cualquier forma binaria real dada, cuya longitud sea como máximo el grado de dicha forma (capítulo 2). Conocida la cota anterior, adaptamos el algoritmo de Sylvester (resultado obtenido en el s. XIX para el caso de formas binarias complejas) al caso real, con el fin de determinar una WD con longitud mínima, es decir, la que da el rango. Usamos matrices bezoutianas para lograr esta descomposición óptima. Este resultado se encuentra en el capítulo 3...Waring's Problem over polynomial rings deals with the problem of decomposing a homogeneous polynomial of degree d as a linear combination of dth powers of linear forms. The goal of this work is to study this problem in the case of real homogeneous polynomials in two variables or real binary forms. One of the aspects related to Waring's Problem is to determine the length of these decompositions or to bound it. Our work proposes (Chapter 2) a constructive method to obtain a real Waring decomposition (WD) for any given real binary form, whose length is at most the degree of that form. In Chapter 3, knowing the previous boundary, we adapt Sylvester's Algorithm (result obtained in the XIX century for the case of complex binary forms) to the real case, in order to determine a WD with minimum length, that is, the one that gives the rank. We use Bezoutian matrices to achieve this optimal decomposition...Fac. de Ciencias MatemáticasTRUEunpu

A Note on Tropical Triangles in the Plane

We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.MTMUCMDepto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu