From the finite to the infinite

Harry Gulliver (The University of Manchester)


The Ax-Grothendieck theorem says (roughly) that for any n polynomials \(f_1,…,f_n\) in n variables with complex coefficients, if the map \((f_1,…,f_n):\mathbb{C}^n\to\mathbb{C}^n\) is injective, then it is surjective. This result has a very surprising and elegant proof, which begins by proving the statement for finite fields (where it’s absolutely trivial and has nothing to do with the field structure or the maps being polynomials), and then levering it up to prove it for \(\mathbb{C}\) (and, in fact, all algebraically closed fields). This is remarkable, because there are no ring maps between \(\mathbb{C}\) and any finite field – algebraically, they have nothing to do with one another, so it’s very surprising to be able to transfer information from the finite case to characteristic zero. This talk will present the ideas behind this proof and explain how this transfer of information takes place.


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