Difference varieties
and algebraic varieties
(Un exposé de Ehud Hrushovski au séminaire de
théorie des nombres de Chevaleret le 20 février 2006)
Résumé :
A difference equation is a polynomial equation involving also a
symbol $\si$ for a ring endomorphism.
Classically, difference equations arose as discrete analogs of
differential equations, with $f^\si(x) = f(x+1)$ for $f$ in some
function field. An example over $\Zz$ is $X^{\si^2} X^6 = X^{5 \si}
$; the solution set in $(\bar{\Qq},\si)$ is a group of roots of
unity
depending on an automorphism $\si$ of $\bar{\Qq}$ (all roots of unity
for certain $\si$.) I will survey some of the model
theory of difference varieties, with emphasis on their arithmetic nature.