Hilbert's Tenth Problem Today: Main Results and Open Problems

Among the 23 problems stated by David Hilbert in 1900 we find:

10. Determination of the Solvability of a Diophantine Equation. Given
a diophantine equation with any number of unknown quantities and with
rational integral numerical coefficients: * To devise a process
according to which it can be determined by a finite number of operations
whether the equation is solvable in rational integers.*

The problem was shown to be undecidable in 1970. Since that time over 300 papers have been published about simplifications, improvements, and applications of this result in different branches of mathematics. In the lecture I plan to survey the main achievements in this area and discuss some important related questions which still remain open.

Last modified: Wed Feb 2 10:48:29 PST 2000