Formerly I was interested in Real
Algebraic Geometry. My Master thesis (UCM,
1984, advisor: J.M.
Sancho) dealt with
Zariski-dense curves. In my Ph. D. thesis (Stanford University,
1988, advisor: G.W. Brumfiel),
I defined and studied spaces
of valuations on rings
compatible with orderings (I called these spaces "real Riemann
surfaces", following denotation by O. Zariski) and spaces of orderings
together with involutions (called "the
complex spectrum of a ring"), similar to the real spectrum of M. Coste
and M. F. Roy. I tried to put these spaces in
connection with several compactifications of algebraic varieties known
then (due to G. Bergman, and to J. Morgan and P. Shalen). Later on, I
studied atypical values of real polynomial functions on
the plane. Later, I pursued the task of making a few facts about real
algebraic curves (well--known
to experts) available to a wider audience.
See my list
of related published papers:
Multiplicidad de intersección y
resultantes. (Spanish) Mathematical contributions:
volume in honor of Professor Enrique Outerelo Domínguez
(Spanish), 333--348, Homen.
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)
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)
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
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
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
solved in full detail. Also, I stress some techniques to draw algebraic
curves. Here are some misprints