Computational aspects of the diophantine problems

Objective

In my thesis, I have studied several diophantine problems and conjectur of Number Theory. In particular, the computational aspect of the abc and the Szpiro conjectures and their applications were extensively explored. I presented several algorithms to test these conjectures leading to the best examples currently known.
In future, I propose to continue to study some conjectures and diophantine problems in Number Theory. I wish to go deeper into four of them:
1. The abc conjecture for number fields. I wish to make explicit several parameters for it.
2. Links between minimal discriminant and conductor of hyperelliptic curves. The Szpiro's conjecture was extended to hyperelliptic curves by P. Lockhart. It remains to specify some parameters in this conjecture and to study its consequences.
3. Finding algebraic curves with several rational points.
4. Studying diophantine problems in number fields.
This work can be done using the computer algebra system SIMATH, develop by the research group of Prof. H.G. Zimmer at the university of Saarbruecken.

Programme(s)

• FP4-TMR - Specific research and technological development programme in the field of the training and mobility of researchers, 1994-1998

Topic(s)

• 0302 - Post-doctoral research training grants
• TM21 - Algebra and Number Theory

Funding Scheme

Coordinator

Participants (1)