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CHRISTOPHE DOCHE
Lecturer à l'université de Macquarie (Sydney)
Division of ICS
Building E6A Office 376
Macquarie University
NSW 2109 Australia
Phone: (+61) 2 9850 9576
Fax: (+61) 2 9850 9551
email: doche_at_ics.mq.edu.au
Curriculum vitae Research Collective activities Talks Teaching Links
Curriculum vitae
Born June 25, 1971 in Bordeaux.
Degrees:
Baccalauréat C (Bordeaux, 1989)
Deug de Droit (University Bordeaux I, 1992)
Deug A (University Bordeaux I, 1993)
Licence de Mathématiques Pures (University Bordeaux I, 1994)
Maîtrise de Mathématiques Pures (University Bordeaux I, 1995)
DEA de Mathématiques Pures (University Bordeaux I, 1996)
Agrégation de Mathématiques (option Mathématiques pour l'Informatique, 1996)
Doctorat de Mathématiques Pures (University Bordeaux I, 2000)
Positions:
Ph.D. student from 1997 to 2000.
Moniteur at the University Bordeaux I from 1998 to 2000 (lectures at the IUT A of Bordeaux).
A.T.E.R. at the University Bordeaux I from 2000 to 2001.
Cryptograher for the AREHCC project from 2001 to 2003.
Lecturer at Macquarie University from 2003.
Research
I obtained a D.E.A. (master) in number theory at the University Bordeaux I in 1996, supervised by Michel Langevin.
My report entitled Problème de Lehmer, nombres de Pisot et de Salem au fil des travaux de Boyd, can be found here: dea.ps.gz (zipped postscript format. Using Windows you will need gzip.exe and ghostview to view this file).I defended my thesis entitled Mesures de Mahler et racines réelles de certaines familles de polynômes on June 21, 2000 at the University Bordeaux I [these.ps.gz]. My Ph.D. advisors were Laurent Habsieger and Michel Langevin.
I am interested in analytic number theory and algorithmic. For example, you can get a file of routines PARI [mahler.gp] to compute particularly Mahler measures of polynomials in one and two variables. More specifically I am working on Lehmer's problem and on the determination of the number of real zeros of certain families of polynomials with coefficients plus or minus 1, e.g. Thue-Morse polynomials.
I explore also another topic with Laurent Habsieger, namely the Rudin-Shapiro polynomials and more precisely the moments of order q of these polynomials. The only result previously known, due to Littlewood, gives the recurrence satisfied by these moments for q=4. We were able to determine explicitly similar linear recurrences for all the even moments q between 4 and 32. Furthermore if L is the length of the Rudin-Shapiro polynomial of order n, that is the sum of the absolute value of its coefficients i.e. L=2^n, we showed that the asymptotic behaviour of these moments was (2L)^(q/2)/(q/2+1) for all even q less than or equal to 52. This result was expected and conjectured by Saffari. We try to extend this work to all q and to generalized Rudin-Shapiro polynomials.
Finally I am doing research on efficiency aspects of cryptosystems based on elliptic and hyperelliptic curves defined over finite fields.
My list of papers can be found below:
On the Zhang Zagier height:
On the spectrum of the Zhang-Zagier height, Math. Comp. 70 (2001), no. 233, 419-430. [spectrum.ps.gz]
Zhang-Zagier heights of perturbed polynomials, Journal de Théorie des Nombres de Bordeaux 13 (2001), 103-110, proceedings des XXIe Journées Arithmétiques. [perturbed.ps.gz] Sur les zéros réels des polynômes :
On the real roots of generalized Thue-Morse polynomials, Acta Arith. XCIX.4 (2001), 309-319. [genethuemorse.ps.gz] Integral Geometry and real zeros of Thue-Morse polynomials, with Michel Mendès France, Experiment. Math. 9 (2000), no. 3, 339-350. [thuemorse.ps.gz]
Real roots! with Michel Mendès France, Analytic and probabilistic methods in number theory (Palanga), TEV, Vilnius 2002. [realroots.ps.gz] An exercise on the average number of real zeros of random real polynomials with Michel Mendès France, Finite and infinite combinatorics (Budapest 2001). J. Bolyai Mathematical Society. [finf.ps.gz]
On the moments of the Rudin-Shapiro polynomials :
Moments of the Rudin-Shapiro Polynomials, with Laurent Habsieger, to appear in the Journal of Fourier Analysis and Applications. [049.ps.gz] Even moments of generalized Rudin--Shapiro polynomials, submitted to Math. Comp. [genersmc.ps.gz] A gp program to precisely compute the moments of the Rudin-Shapiro polynomial of order q even less than or equal to 32 and the moments of some generalized Rudin-Shapiro polynomials is available here [moments.gp.gz] as well as the tables of minimal polynomials for the recurrences and the first moments [data.gz] et [data32.gz]
On an efficient implementation of finite fields in characteristic 2:
Redundant Trinomials for Finite Fields of Characteristic 2, preprint. [redundant.ps.gz] A table of redundant polynomials written in gp is also available.
Collective activities
Organizer with Jean-Jacques Ruch of the Séminaire de Théorie Analytique des Nombres de Bordeaux since 1999.
Organizer with Jean-Paul Allouche and Jean-Jacques Ruch of the Colloque en l'honneur de Michel Mendès France from
September 11 to 14, 2000 in Bordeaux.
Talks
On the spectrum of the Zhang-Zagier height:
Groupe de Travail de Théorie Analytique des Nombres, Avril 1998, Bordeaux.
Colloque Jeunes Chercheurs, September 1998, Caen.
Groupe d'Etude des Problèmes Diophantiens, December 1998, Paris.
Séminaire de Théorie des Nombres, February 1999, Metz.
XXIe Journées Arithmétiques, July 1999, Roma.
Séminaire Dynamique Arithmétique Combinatoire, December 2000, Luminy.
FINF conference for the 70th birthday of A. Hajnal and V. T. Sós, January 2001, Budapest.
Journées Nombres, Suites,... April 2001, Saint-Etienne.
Number theory seminar, november 2001, Basel.On the real roots of Thue-Morse polynomials:
Groupe de Travail de Théorie Analytique des Nombres, March 1999, Bordeaux.
Colloque Jeunes Chercheurs, September 1999, Lyon.
Journée Metz Strasbourg Freibourg, May 2000, Metz.
Colloque en l'honneur de Michel Mendès France, September 2000, Bordeaux.
Séminaire de Théorie analytique et élémentaire des nombres, March 2001, Paris.
On cryptography:
Problèmes d'arithmétique en cryptographie, talk given during the journées de l'IREM d'aquitaine, Jun. 2002, Bordeaux.
La cryptographie : le message secret à travers les âges, conferences given during la fête de la science , Oct. 2002, Ajaccio, Bastia, Corte.
Teaching
Agrégation de Mathématiques (option Mathématiques discrètes, 1996).
First degree lectures since 1998 at the IUT A (computer science) of Bordeaux.
First degree lectures since 2000 at the University Bordeaux I.
First degree lectures since 2000 at the University of Agen.
COMP238 and COMP115 at Macquarie University. at Macquarie University.
Links
PARI.
Wolfram Mathematics encyclopaedia.
TeXPoint: enables to include LaTeX formulae in Powerpoint presentations.
Lehmer's problem.
Heights of Polynomials and Entropy in Algebraic Dynamics by G. Everest et T. Ward.
Peter Borwein's homepage.
David Boyd's homepage.
Chris Smyth's homepage.
Sloane's On-Line Encyclopedia of Integer Sequences.
Virtual Rubik's cube and how to solve it.
View of earth and of Europe at this moment.
Some of the best advertisementsLast modified March, 4th 2004.
