Frontiers in Mathematical Sciences

Logical Arithmetic and Real Algebraic Geometry

Shahram Mohsenipour


We will talk about the diophantine equations from a logical point of view and explain that for a subsystem $T$ of the Peano Arithmetic how diophantine equations can be studied within $T$. For a special $T$ called Open Induction, this study leads us, through Wilkie's theorem, to an algebro-geometric situation wherein we consider a kind of "arithmetic" solutions of diophantine equations in the space of all solutions. This latter space is the real spectrum of a commutative ring. In modern real algebraic geometry, the real spectrum has the importance of the Zariski spectrum in the usual algebraic geometry. We will also briefly discuss a recent result of the speaker on the distribution of the "arithmetic" points in the real spectrum.