Search

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 24, No. 1, pp. 5–30, 2022., Square matrices A and B are orthogonal if AʘB = Z = BʘA, where Z is the matrix entries equal to 0, and ʘ is the tropical matrix multiplication. We study orthogonality for normal matrices over the set {0, −1}, endowed with tropical addition and multiplication. To do this, we investigate the orthogonal set of a matrix A, i.e., the set of all matrices orthogonal to A. In particular, we study the family of minimal elements inside the orthogonal set, called a basis. Orthogonal sets and bases are computed for various matrices and matrix sets. Matrices whose bases are singletons are characterized. Orthogonality and minimal orthogonality are described in the language of graphs. The geometric interpretation of the results obtained is discussed., Ministerio de Economía y Competitividad (MINECO), Ministerio de Ciencia e Innovación (MICINN), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, FALSE, pub

Let a polynomial function f of two real variables be given. We prove the existence of a finite number of unbounded regions of the real plane along which the tangent planes to the graph of f tend to horizontal position, when moving away from the origin. The real limit values of this function on these regions are called asymptotic values. We also define the real critical values at infinity of f and prove the theorem of local trivial fibration at infinity, away from these values., D.G.I.C.Y.T., C.I.C.Y.T., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi-definite nondefinite or definite. We present a discussion about isolated points. By means of the P operator, these points can be easily identified for curves defined by minimal polynomials of order bigger than one. We also discuss the conditions that a curve must satisfy in order to have a minimal polynomial. Finally, we list the most relevant topological properties of affine and projective, complex and real plane algebraic curves., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We correct a previously published theorem by proving a criterion to decide whether the fibration given by a real polynomial function f : R-2 --> R is locally trivial at infinity. The algorithmical nature of this criterion provides a test that can be applied to any real polynomial f., DGICYT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonalityequivalence relation are computed., Ministerio de Economía y Competitividad, Universidad Complutense de Madrid, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we fully describe all tropical linear maps in the tropical projective plane, that is, maps from the tropical plane to itself given by tropical multiplication by a real 3×3 matrix A. The map fA is continuous and piecewise-linear in the classical sense. In some particular cases, the map fA is a parallel projection onto the set spanned by the columns of A. In the general case, after a change of coordinates, the map collapses at most three regions of the plane onto certain segments, called antennas, and is a parallel projection elsewhere (Theorem 3). In order to study fA, we may assume that A is normal, i.e., I A 0, up to changes of coordinates. A given matrix A admits infinitely many normalizations. Our approach is to define and compute a unique normalization for A (which we call lower canonical normalization) (Theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning. On , any , some aspects of tropical linear maps have been studied in [6]. We work in , adding a geometric view and doing everything explicitly. We give precise pictures. Inspiration for this paper comes from [3,5,6,8,12,26]. We have tried to make it self-contained. Our preparatory results present noticeable relationships between the algebraic properties of a given matrix A (idempotent normal matrix, permutation matrix, etc.) and classical geometric properties of the points spanned by the columns of A (classical convexity and others); see Theorem 2 and Corollary 1. As a by-product, we compute all the tropical square roots of normal matrices of a certain type; see Corollary 4. This is, perhaps, a curious result in tropical algebra. Our final aim is, however, to give a precise description of the map . This is particularly easy when two tropical triangles arising from A (denoted and ) fit as much as possible. Then the action of fA is easily described on (the closure of) each cell of the cell decomposition ; see Theorem 3. Normal matrices play a cr, UCM, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

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.

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., MTM, UCM, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

This paper provides an elementary exposition of affine and projective real plane curves. Where possible, elementary proofs are given, but to some extent, one must make use of results from algebraic geometry (over the complex numbers) and real algebraic geometry. The bulk of the paper discusses algebraic aspects of affine and projective curves, but there is a short section at the end on topological aspects. A major theme of the paper is that real affine curves VR(f) with f an indefinite polynomial in R[X, Y ] have similar properties to complex affine curves, but semidefinite polynomials give rise to much different behavior. Many examples are given to illustrate the concepts., DGICYT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We correct a previously published theorem by proving a criterion to decide whether the fibration given by a real polynomial function f : R-2 --> R is locally trivial at infinity. The algorithmical nature of this criterion provides a test that can be applied to any real polynomial f., DGICYT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The Baer-Krull theorems deal with the relationship between orderings of a valued field compatible with the valuation, and orderings of the residue class field. For these theorems it is necessary that the valuation ring should be convex with respect to the ordering. For a real field R, and an extension K _ R, the author defines SpecC(K/R) in terms of equivalence classes of embeddings of K over R into an algebraic closure C of K. This is done in such a way that when K is also real, the points of SpecC(K/R) correspond to the orderings of K over R. Given a point of SpecC(K/R), the author extends the definition of convexity to subsets of K (again this is the usual definition when K is real). Now let R be real, K be an extension of R, and B be a valuation ring in K. Let R be the residue class field of R\B. Suppose that R is a real subfield of K. The author studies relations between SpecC(K/R) and SpecC(K/R). In particular, it is shown that there is a lifting of each element of SpecC(K/R) to an element of SpecC(K/R), compatible with the valuation, and such that the lifting has the generalised convexity property. While a more elementary treatment of this result is possible if R = Q, for general R the proof involves model theory in a nontrivial way., CICYT, DGICYT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let a polynomial function f of two real variables be given. We prove the existence of a finite number of unbounded regions of the real plane along which the tangent planes to the graph of f tend to horizontal position, when moving away from the origin. The real limit values of this function on these regions are called asymptotic values. We also define the real critical values at infinity of f and prove the theorem of local trivial fibration at infinity, away from these values., D.G.I.C.Y.T., C.I.C.Y.T., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Let f : R(2) --> R be an open polynomial function. Then, f changes sign across V(f) (alternatively around a singular point of V(f)) and the function c : R --> N expressing the number c(lambda) of connected components of the lambda-level curve of f is lower semicontinuous; it has removable singularity at every value lambda which is critical and is not a real critical value at infinity for f., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

“Let A be the coordinate ring of the real plane, i.e., A = R[X, Y ], where X, Y are algebraically independent over the field of real numbers R. The purpose of this paper is to describe all the valuation rings of A which are compatible with some prime cone of A, i.e., to give a description of all the points of the real Riemann surface Sr(A/R). Furthermore, we explain the geometric meaning of these points and indicate the specialization relations among them.”