My
current research interest is Tropical
Algebra and
Geometry.

See my research papers (some on arXiv) regarding tropical issues:

See my research papers (some on arXiv) regarding tropical issues:

- (with J. Linde) Matrices
commuting
with a given normal tropical matrix,
Linear Algebra and Its Applications (2015), pp. 101-121, DOI information: 10.1016/j.laa.2015.04.032, also found in ArXiv 1209.0660

- Distances
on
the
tropical line determined by two points Kybernetika, 50, n. 3, (2014), 408-435, ArXiv
1310.0174,

- On tropical Kleene star matrices and alcoved polytopes, Kybernetika, 49, n.6, (2013), 897-910, ArXiv 1210.1735
- (with A. Jiménez) Characterizing the
convexity of
the n-dimensional tropical simplex and the six convex
classes in R^3,
(2011) ArXiv 1205.4162

- (with E. Lorenzo) An algortihm to solve any tropical linear system A x=B x, Linear Algebra and its Applications, 435, n.4, (2011), 884--901;
- Tropical linear maps on the plane, Linear Algebra and its Applications, 435, n.7, (2011), 1681--1710; doi:10.1016/j.laa.2010.07.031 (2010), MR2810666 (2012i:14075)
- (with M. Ansola) Tropical
conics for the layman, in Tropical and Idempotent
Mathematics, in
G.L. Litvinov and S. N. Sergeev (eds.), Contemp. Math. 495
(2009) 87-101

- (with M. Ansola) A note on tropical triangles in the plane, in Acta Math. Sinica, Engl. v. 25, no. 11, (2009) 1775-1786

Here is a pdf of my talk
on Matrices commuting with a given normal tropical
matrix, given at Skolkovo, near Moscow, in August 2015.

- Cómo obtener curvas con formas predeterminadas a partir de circunferencias (2010), ArXiv 1307.7431
- Real plane algebraic curves.
*Expo. Math.*20 (2002), no. 4, 291--314. 14P05,**MR1940009****(2003h:14088)** - Multiplicidad de intersección y
resultantes. (Spanish)
*Mathematical contributions: volume in honor of Professor Enrique Outerelo Domínguez (Spanish),*333--348, Homen. Univ. Complut.,*Editorial Complutense, Madrid,*2004. 14H50 (13P99),**MR2212975****(2006k:14052)** - Curvas algebraicas reales planas. (Spanish)
*Mathematical contributions: volume in honor of Professor Joaquín Arregui Fernández (Spanish),*249--263, Homen. Univ. Complut.,*Editorial Complutense, Madrid,*2000. 14P05, MR 1803907 - (with M.
Coste) Atypical
values at infinity of a
polynomial function on the real plane: an erratum, and an
algorithmic
criterion.
*J. Pure Appl. Algebra*162 (2001), no. 1, 23--35. 14P25 (14P20 26C99),**MR1844807**(2002f:14078) - (with J.
Ferrera) The asymptotic
values of a
polynomial function on the real plane.
*J. Pure Appl. Algebra*106 (1996), no. 3, 263--273. 26C99 (14P99),**MR1298760****(95j:14076)** - (with J.
Ferrera)
Level curves of open polynomial functions
on the real
plane.
*Comm. Algebra*22 (1994), no. 14, 5973--5981. 14P25 (26C05),**MR1298760****(95j:14076)** - A complex version of the Baer-Krull
theorems.
*Comm. Algebra*28 (2000), no. 8, 3727--3737. 12J15 (12J10 12L12),**MR1767584**(2001i:12010) - The
complex spectrum of a ring.
*Real algebraic geometry and ordered structures (Baton Rouge, LA, 1996),*235--249, Contemp. Math., 253,*Amer. Math. Soc., Providence, RI,*2000. 13J30 (12D15),**MR1747588****(2001f:13039)** - Specializations and a local
homeomorphism
theorem for real Riemann surfaces of rings.
*Pacific J. Math.*176 (1996), no. 2, 427--442. 14P10 (12D15),**MR1435000****(98b:14044)** - The compatible valuation rings of the
coordinate ring of the real plane.
*Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991),*231--242, Contemp. Math., 155,*Amer. Math. Soc., Providence, RI,*1994. 13F30 (13A18),**MR1260710****(94k:13029)**

Prior to getting
interested in Tropical Mathematics, I devoted some
years to write a text book (undergraduate
level) on plane
algebraic curves (complex and
real). It is written in Spanish. About half of the
exercises in
it are
solved in full detail. Also, I stress some techniques to draw
algebraic
curves. Here are some misprints
and errors