Proud survivor of the overly selected, supposedly elitist and somehow traumatized, 1st Promotion of Mathematicians ever formed in Cantabria (October 1979-June 1984). I did my PhD Thesis in Mathematics under the smart advising of Tomás Recio , defended in February 1987. The same year I became Associate Professor (Profesor Titular) of "Algebra" (25 years old). In January 1988, I had to leave the the University, returning to the academic system in February 1989. I returned with the firm conviction of devoting most of my time and effort to the scientific formation of young mathematicians through a high quality doctoral program in Mathematics at the University of Cantabria (the program does not exist anymore). I was advising my first PhD Student (J. L. Montaña ) when I was 27 years old. Since then I have tried to interpret my research as a pathfinder, seeking less traveled roads in the frontiers of mathematical knowledge, accompanied by a PhD Student who gets thus a formation as researcher.

Research Interests

Most of my research interests belong to the field of Computational Mathematics. I focussed on Computational Complexity , Foundations of Computational Mathematics, either Symbolic or Numeric Effective Methods in Algebraic Geometry. Obviously, these interests also include related fields as Commutative Algebra, Algebraic, Arithmetic and Diophantine Geometry, Numerical Analysis, Linear Algebra, Theoretical Computer Science, Integral Geometry and many others. In my poor opinion, there is no isolated field of knowledge in Mathematics, there is just a missed path which connects it to the others .


Some interesting moments of my scientific career include the TERA Project (which I co-directed with Joos Heintz and Marc Giusti ) in the nineties of the past century. The manuscripts we published (together with my former student J.E. Morais) in this period yielded the most efficient symbolic method that solves universally systems of multivariate polynomial equations. Another milestone was the solution of Smale's 17th Problem toghether with my former student C. Beltrán , obtained after several joint publications in the first decade of the present century. Other milestones were the works on the Arithmetic Nullstellensatz ( firstly developped with Teresa Krick , and, later on, also with M. Sombra and my former student K. Hägele ). Last but not least, I have devoted an special effort to understand lower complexity bounds of several problems in several contexts as Algebraic Complexity Theory (including a solution to M. Ben-Or's conjeture on the usage of higher order Betti numbers as lower complexity bound, published with J.E. Morais and J.L. Montaña) or to prove exponential lower complexity bounds for universal, geometrically robust algorithms for eliminating a single block of existencial quantifiers (most of them with J. Heintz and G. Matera ).

Research Production

I have published more than 60 scientific manuscripts. Some of them have been well treated by my colleagues, yielding an h-index 18 in Thomson Reuters Web of Science (with more than 800 citations), and an h-index 21 in Google Scholar (with more than 1900 citations in this data basis) and an h-index 21 in Research Gate . I have completed 30 years of positive evaluation ( 5 periods of 6 years each, called "sexenios" in the Spanish academic system). I should apply for next evaluation on December, 2023 if not in retirement.

I also liked Editing, being a member of the Editorial Board of the Journal of Complexity since 2004 and of the Editorial Board of Applicable Algebra in Engineering Communications and Computing since 2014. I have also edited several books and Special Issues of Journals.

I have leaded several national Research Grants and participated in many since my youngest years. I also organized (either in the scientific committe or in the local committee) scientific meetings and conferences . Colleagues found interesting to invite me to give several plenary and invited talks and I presented a few referred communications along these years. I have also been invited to visit several scientific Laboratories and Universities.


I have already completed 30 years of positive evaluation in teaching ( 6 periods of 5 yers each, called "quinquenios" in the Spanish academic system) (December 31st, 2014). This is the maximum number of evaluable periods of teaching, although it does not mean that you don't teach anymore at Spanish universities. It simply means that you may avoid any future teaching evaluation, if you wish. In all these years, I have taught many courses either of undergraduate or graduate programs. I have devoted more than 5200 hours lecturing in a classroom , not counting the preparation, correction of excercises and projects, exams or tutorial hours, which, obviously, were much more than these. Among undergraduate courses, I was committed with courses in Mathematics Studies (as Commutative Algebra, Linear Algebra, Computer Algebra, Algebraic Geometry, Geometry or General Topology), Computer Engineering (as Introduction to Logic for Computer Scientists or Automata and Formal Languages Theory), Physics (Linear Algebra) or Nautic Studies (Linear Algebra and Spheric Trigonometry). Among graduate courses I was committed with several courses in the Master Program of "Mathematics & Computation" as Computational Complexity. Effective Methods in Algebraic Geometry, since its occurence in 2008. In former Doctoral Program in Mathematics courses I taught many different courses as Introduction to Computational Complexity, Real Algebraic Geometry, Algebraic Complety Theory, Algorithmic Algebra and others.

Duties & Service to the Community

I have also assumed several duties and services to the Academic Community both in different structures of the University of Cantabria, in national and internatonal Scientific Committees or as chosen delegate of the union CC.OO. and member of its Executive Committee in Cantabria. I'm also the head of the Research Group M&C:FLAI (Mathematics & Computation: Foundations, Learning and Artificial Intelligence).