An extremely short CV
I obtained my Ph.D. at the university of Ghent with a thesis on Hilbert's Tenth Problem. I spent one year at the Departement of Mathematics, University College Dublin as postdoc and two years at the Equipe de Logique Mathematique, University of Paris 7 as a postdoc.
Since 2003 I'm back in Belgium and have been employed at several institutions in very temporary teaching jobs.
Research interests
I am mainly interested in decidablity questions in number theory and algebra. In particular decision problems concerning the solvability of diophantine equations.
I was a supervisor of Jeroen Demeyer's Ph.D.-thesis on Hilbert's Tenth Problem. Jeroen defended his thesis on March 28th, 2007 at the University of Ghent.
Furthermore my research interests include the model theory of fields, and more generally applications of logic to mathematics and vice versa.
Discussing with Bjorn Poonen and Thomas Scanlon at Oberwolfach.
Publications
Cornelissen, Gunther; Zahidi, Karim. Elliptic divisibility sequences and undecidable problems about rational points, to appear in Journal für die Reine und Angewandte Mathematik, complQ.pdf
Pheidas, Thanases; Zahidi, Karim. Decision problems in algebra and analogues of Hilbert's Tenth Problem, to appear in London Mathematical Society Lecture Notes Series (Cambridge University Press) 2006, PheidasZahidi.pdf
Cornelissen, Gunther; Pheidas, Thanases; Zahidi, Karim. Division-ample sets and the Diophantine problem for rings of integers. J. Théor. Nombres Bordeaux 17 (2005), no. 3, 727—735. JNTB.pdf
Zahidi, Karim. On the u-invariant of p-adic function fields. Comm. Algebra 33 (2005), no. 7, 2307—2314. uinv.pdf
Pheidas, Thanases; Zahidi, Karim. Elimination theory for addition and the Frobenius map in polynomial rings. J. Symbolic Logic 69 (2004), no. 4, 1006—1026.
Zahidi, Karim. Hilbert's tenth problem for rings of rational functions. Notre Dame J. Formal Logic 43 (2002), no. 3, 181—192 (2003).
Cornelissen, Gunther; Zahidi, Karim. Topology of Diophantine sets: remarks on Mazur's conjectures. Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), 253—260, Contemp. Math., 270, Amer. Math. Soc., Providence, RI, 2000.
Pheidas, Thanases; Zahidi, Karim. Undecidability of existential theories of rings and fields: a survey. Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), 49—105, Contemp. Math., 270, Amer. Math. Soc., Providence, RI, 2000.
Zahidi, Karim. The existential theory of real hyperelliptic function fields. J. Algebra 233 (2000), no. 1, 65—86.
Zahidi, Karim. On Diophantine sets over polynomial rings. Proc. Amer. Math. Soc. 128 (2000), no. 3, 877—884.
Pheidas, Thanases; Zahidi, Karim. Undecidable existential theories of polynomial rings and function fields. Comm. Algebra 27 (1999), no. 10, 4993—5010.
Preprints
Elliptic divisibility sequences and undecidable problems about rational points
Julia Robinson proved that the exists-forall-exists-forall-exists-theory of the field of rational numbers is undecidable.
In this paper we prove that a certain conjecture about rational points on elliptic curves implies that the universal-existential-theory (i.e. the forall-exists-theory) of the field of rational numbers is undecidable. Accepted for publication by Crelle's journal. (joint work with G. Cornelissen)
complQ.pdf
Decision problems in algebra and analogues of Hilbert's Tenth Problem
This is the text of a tutorial presented at the American Institute of Mathematics (March 2005) and the Newton Institute Cambridge (April 2005). To be published in the proceedings of the Model Theory Workshop in Cambridge (Oxford University Press). Joint work with Thanases Pheidas.
PheidasZahidi.pdf