|Christiaan van de Woestijne||
Nederlands / English
I'm working in number theory, with special interest in (varieties over) finite fields, symbolic computation and complexity of algorithms, and combinatorics.
Since June 2006, I've been attached to an Austrian Science Fund (FWF) project about the application of numeration systems in cryptography. I'm currently working on algebraic and other structural aspects of numeration systems, such as the underlying group and module structure, connection with tilings, and the problem of finding good digit sets for a given base.
My Ph.D. thesis is about deterministic algorithms for solving polynomial equations in many variables over finite fields. My advisor was Prof. H.W. Lenstra, Jr.. Click here for the final version (in PDF). An extended abstract appeared in the Proceedings of ISSAC 2005; click here for the text.
An implementation of the algorithms developed in my thesis is also available: it's here (in MAGMA).
PhD. defense was on Tuesday, 16 May 2006. Photos are here.
Here is a recent CV.
In 1997/98, I did a research project for obtaining my Master's degree in mathematics under supervision of Dr Benne de Weger and Prof. R. Tijdeman. It was a number-theoretic project about almost-powers (integral numbers of the form axk, where a is relatively small); the question was how close we can get such numbers to each other in various constellations. My thesis (in gzipped PostScipt) is available here. In a shorter version, it appeared in two parts in the 1999 volume of Acta Arithmetica (vol. 90).
In 1998/99, I did the same for my Computer Science degree, supervised this time by Dr Hendrik-Jan Hoogeboom, and Dr Crit Cremers from the department of General Linguistics. This time, I was occupied with the Delilah system, developed by Cremers and Hijzelendoorn, which is a parsing system for the written Dutch language. The problem was firstly to relate the Delilah system to grammar formalisms currently studied in theoretical computer science, and secondly to estimate the time and space complexity of the parsing system. You can find my results in my thesis (also in gzipped PostScript).
1. On the diameter of sets of almost powers. Acta Arith. XC.4 (1999), 371-385. With B.M.M. de Weger.
2. On the power-free parts of consecutive integers. Acta Arith. XC.4 (1999), 387-395. With B.M.M. de Weger.
3. Deterministic equation solving over finite fields. Proc. ISSAC 05, Beijing, China, ACM press (2005), 348-353.
4. Deterministic equation solving over finite fields. Ph.D. Thesis Universiteit Leiden, 2006. Available at the DSpace repository in Leiden.
5. Surface Parametrisation without Diagonalisation. Proc. ISSAC 06, Genova, Italy, ACM Press (2006), 340-344. See here.
6. Construction of Rational Points on Elliptic Curves over Finite Fields. Proc. ANTS-VII, Berlin, Germany, edited as LNCS 4076, Springer (2006), 510-524. With A. Shallue.
7. The Casas-Alvero Conjecture for Infinitely Many Degrees. J. Algebra 316 (2007), 224--230. With H.-C. Graf v. Bothmer, O. Labs, and J. Schicho. Available on arXiv in an expanded version (math.AC/0605090).
8. Noncanonical number systems in the integers. J. Num. Th. 128 (2008), 2914-2938. Available on arXiv (0804.2190).
9. The structure of abelian groups supporting a number system (extended abstract). Actes des rencontres du CIRM Vol. 1 no. 1 (2009), p. 75-79. Available here.
10. Exact values for the Waring function. Acta Arith. 141.2 (2010), 171-190. Available on arXiv (0810.0485). With A. Winterhof.
11. Digit systems over commutative rings. With Klaus Scheicher, Paul Surer, and Jörg Thuswaldner. Submitted.
12. Factors of disconnected graphs and polynomials with nonnegative integer coefficients. Ars Mathematica Contemporanea 5 (2012), 307--323. Also available online.
13. A fascinating polynomial sequence arising from an electrostatics problem on the sphere. Acta Mathematica Hungarica 137 (2012), 10--26. Also available online. With Johann S. Brauchart, Peter D. Dragnev, and Ed B. Saff.
14. Number systems and the Chinese Remainder Theorem. Submitted.
15. On the computation of expanding polynomials. In preparation.
16. Number systems in abelian groups. With Ryotaro Okazaki. In preparation.
17. Binary number systems. In preparation.
1. Implementation of the algorithms given in my Ph.D. thesis, in MAGMA.
2. Implementation of the algorithm from item 6. above, for odd characteristic, in MAGMA.
3. KASH 2.x code for constructing admissible vectors, for use with item 9. above.
4. Stuff about expanding polynomials.
Deterministic equation solving in finite fields, Oberwolfach workshop on Finite Fields, December 2004 (PDF).
Deterministic equation solving over finite fields (other version), Algebraic Geometry Seminar, RICAM Linz, and Zahlentheoretisches Kolloquium, TU Graz, November 2005 (PDF).
Surface Parametrisation without Diagonalisation, ISSAC 2006, Genova, Italy, July 2006 (PDF).
Finding Points on Elliptic Curves in Deterministic Polynomial Time (odd characteristic), ANTS-VII, Berlin, Germany, July 2006 (PDF).
Representations of numbers without the digit zero, Cryptography Seminar, TU Graz, June 2007 (PDF).
Exact values for Waring's problem in finite fields, Journées Arithmétiques 2007, Edinburgh, July 2007 (PDF).
Algebraic aspects of number systems, Journées de Numération 2008, Prague, May 2008 (PDF).
Generalised binary number systems, Mathematisches Seminar, Montanuniversität Leoben, 27 February 2009 (PDF).
The structure of groups supporting a numeration system, Numeration: Mathematics and Computer Science, CIRM, Luminy, March 2009 (PDF).
A second look at binary digits, ÖMG-DMV-Kongress, Graz, September 2009 (PDF).
Canonical number systems and algebraic number theory, Workshop on Digital expansions, Dynamics and Tilings, Aussois, April 2010 (PDF).
Number systems and the Chinese Remainder Theorem (dedicated to Prof. Attila Pethő on occasion of his 60th birthday), Number theory and its applications, Debrecen, 4-8 October 2010.
A second look at binary digits, Mathematical seminar, Hungarian Academy of Sciences, Budapest, 13 October 2010.
Deterministic equation solving over finite fields, Séminaire de Cryptographie, IRMAR, Université Rennes I, 25 March 2011.
Factorisation in monoids of finite graphs, AlgNTH Seminar, Karl-Franzens-Universität Graz, 31 March 2011.
Fibred products of number systems, Numération 2011, Liège, 6-10 June 2011 (PDF).
On the Casas Alvero conjecture, Czech and Slovak International Conference on Number Theory, Stará Lesná, Slovakia, 4-8 September 2011 (PDF).
Exact bounds for Waring's problem with large exponent, Workshop on Finite fields and their applications, Strobl, Austria, 3-7 September 2012.