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M. J. de la Puente

Publications and source records attributed to M. J. de la Puente.

11 recordsLinked to original sources
arXiv2020

Quasi--Euclidean classification of Alcoved Convex Polyhedra

We give the quasi--Euclidean classification of the maximal (with respect to the $f$--vector) alcoved polyhedra. The $f$--vector of these maximal convex bodies is $(20,30,12)$, so they are simple dodecahedra. We find eight quasi--Euclidean classes. This classification, which preserves angles, is finer than the known combinatorial classification (found in 2012 by Jiménez and de la Puente), which has only six classes. Each alcoved polyhedron $\mathcal{P}$ is represented by a unique visualized idempotent matrix $A$. Some 2--minors of $A$ are invariants of $\mathcal{P}$: they are the tropical edge--lengths of $\mathcal{P}$.

math.CO↗
arXiv2015

Matrices de rotaciones, simetr\'ıas y roto-simetr\'ıas

In this note we find the orthogonal matrices $R,S\in M_3(\mathbb{R})$ corresponding to the clockwise rotation $r$ in $\mathbb{R}^3$ around the axis generated by a unit vector $u=(a,b,c)^t$ through an angle $α\in [0,2π)$, and to the symmetry $s$ in $\mathbb{R}^3$ on the plane perpendicular to $u$. Matrix $S$ depends on $a,b,c$ and matrix $R$ depends on $a,b,c, \cos α$ and $\sin α$. We show $SR=RS$. The matrix $R$ is due to Alperin.

math.HO↗
arXiv2014

Matrices commuting with a given normal tropical matrix

Consider the space $M_n^{nor}$ of square normal matrices $X=(x_{ij})$ over $\mathbb{R}\cup\{-\infty\}$, i.e., $-\infty\le x_{ij}\le0$ and $x_{ii}=0$. Endow $M_n^{nor}$ with the tropical sum $\oplus$ and multiplication $\odot$. Fix a real matrix $A\in M_n^{nor}$ and consider the set $Ω(A)$ of matrices in $M_n^{nor}$ which commute with $A$. We prove that $Ω(A)$ is a finite union of alcoved polytopes; in particular, $Ω(A)$ is a finite union of convex sets. The set $Ω^A(A)$ of $X$ such that $A\odot X=X\odot A=A$ is also a finite union of alcoved polytopes. The same is true for the set $Ω'(A)$ of $X$ such that $A\odot X=X\odot A=X$. A topology is given to $M_n^{nor}$. Then, the set $Ω^{A}(A)$ is a neighborhood of the identity matrix $I$. If $A$ is strictly normal, then $Ω'(A)$ is a neighborhood of the zero matrix. In one case, $Ω(A)$ is a neighborhood of $A$. We give an upper bound for the dimension of $Ω'(A)$. We explore the relationship between the polyhedral complexes $span A$, $span X$ and $span (AX)$, when $A$ and $X$ commute. Two matrices, denoted $\underline{A}$ and $\bar{A}$, arise from $A$, in connection with $Ω(A)$. The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.

math.RA↗
arXiv2014

Distances on the tropical line determined by two points

Let $p',q'\in R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $R^n/\sim$ uniquely determine a tropical line $L(p,q)$, passing through them, and stable under small perturbations. This line is a balanced unrooted semi--labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p'$ and $q'$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also prove that $L(p,q)$ is caterpillar. We prove that every vertex in $L(p,q)$ belongs to the tropical linear segment joining $p$ and $q$. A vertex, denoted $pq$, closest (w.r.t tropical distance) to $p$ exists in $L(p,q)$. Same for $q$. The distances between pairs of adjacent vertices in $L(p,q)$ and the distances $\dd(p,pq)$, $\dd(qp,q)$ and $\dd(p,q)$ are certain entries of the matrix $|F|$. In addition, if $p$ and $q$ are generic, then the tree $L(p,q)$ is trivalent. The entries of $F$ are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of $A$.

math.MG↗
arXiv2013

On tropical Kleene star matrices and alcoved polytopes

In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero--diagonal real matrix $A$ is characterized by $A$ being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius of a hyperplane section of its tropical span.

math.MG↗
arXiv2012

Six combinatorial clases of maximal convex tropical polyhedra

In this paper we bring together tropical linear algebra and convex 3-dimensional bodies. We show how certain convex 3-dimensional bodies having 20 vertices and 12 facets can be encoded in a $4\times 4$ integer zero-diagonal matrix $A$. A tropical tetrahedron is the set of points in $\R^3$ tropically spanned by four given tropically non-coplanar points. It is a near-miss Johnson solid. The coordinates of the points are arranged as the columns of a $4\times 4$ real matrix $A$ and the tetrahedron is denoted $\spann(A)$. We study tropical tetrahedra which are convex and maximal, computing the extremals of $\spann(A)$ and the length (tropical or Euclidean) of its edges. Then, we classify convex maximal tropical tetrahedra, combinatorially. There are six classes, up to symmetry and chirality. Only one class contains symmetric solids and only one contains chiral ones. In the way, we show that the combinatorial type of the regular dodecahedron does not occur here. We also prove that convex maximal tropical tetrahedra are not vertex-transitive, in general. We give families of examples, circulant matrices providing examples for two classes. A crucial role is played by the $2\times 2$ minors of $A$.

math.CO↗
arXiv2011

Tropical linear maps on the plane

In this paper we fully describe all tropical linear mappings in the tropical projective plane, that is, maps from the tropical plane to itself given by tropical multiplication by an order 3 matrix. An erratum has been added fixing two mistakes in the main paper.

math.AG↗
arXiv2011

An algorithm to describe the solution set of any tropical linear system $A\odot x=B\odot x$

An algorithm to give an explicit description of all the solutions to any tropical linear system $A\odot x=B\odot x$ is presented. The given system is converted into a finite (rather small) number $p$ of pairs $(S,T)$ of classical linear systems: a system $S$ of equations and a system $T$ of inequalities. The notion, introduced here, that makes $p$ small, is called compatibility. The particular feature of both $S$ and $T$ is that each item (equation or inequality) is bivariate, i.e., it involves exactly two variables; one variable with coefficient $1$, and the other one with $-1$. $S$ is solved by Gaussian elimination. We explain how to solve $T$ by a method similar to Gaussian elimination. To achieve this, we introduce the notion of sub--special matrix. The procedure applied to $T$ is, therefore, called sub--specialization.

math.RA↗
arXiv2008

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 $\infty,0,1$. 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.

math.CO↗
arXiv2008

Tropical conics for the layman

We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining polynomial. Finally, we characterize those degree--two tropical polynomials which are reducible and factorize them. We show that there exist irreducible degree--two tropical polynomials giving rise to pairs of tropical lines.

math.AG↗