What do you do to start reading dimension theory? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this dimension theory.

Aspects of topology: On dimension theory

We prove a general Zariski-van Kampen-Lefschetz type theorem for higher homotopy groups of generic and nongeneric pencils on singular open complex spaces.

The Zariski-Lefschetz principle for higher homotopy groups of nongeneric pencils

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.

/pdf/computing-the-vertices-of-tropical-polyhedra-using-directed-1tho5iwkom.pdf

Computing the Vertices of Tropical Polyhedra Using Directed Hypergraphs

We consider the family of fibres of a polynomial function f on a smooth noncompact algebraic real surface and we characterise the regular fibres of f which are atypical due to their asymptotic behaviour at infinity. We compare to the similar problem in the complex case.

Asymptotic behaviour of families of real curves

Let $f: \mathbb K ^n \rightarrow \mathbb K $ be a polynomial, $\mathbb K =\mathbb R , \,\mathbb C $. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of $f$.

Reaching generalized critical values of a polynomial

https://perso.univ-rennes1.fr/michel.coste/publis/atypical.pdf

Atypical values at infinity of a polynomial function on the real plane: an erratum, and an algorithmic criterion

http://www.mat.ucm.es/~mpuente/pdfs/linde_puente_LAA.pdf

Matrices commuting with a given normal tropical matrix

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 $A$.

Six combinatorial clases of maximal convex tropical polyhedra

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

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.

M.J. de la Puente

Papers

Atypical values at infinity of a polynomial function on the real plane: an erratum, and an algorithmic criterion

Matrices commuting with a given normal tropical matrix

Six combinatorial clases of maximal convex tropical polyhedra

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

On tropical Kleene star matrices and alcoved polytopes