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Nonstandard Algebraic Geometry

Posted: Mon Sep 05, 2016 9:23 pm
by Brian
Does anyone know if anyone is working on this still??

I am trying to prove that x^3 + y^3 = 1 has no rational nontrivial solutions by using nonstandard analysis principles.

The transfer principle says that there are rationals (x,y) in Q^2 that are not trivial and satisfy equation if and only if

there are hyper-rationals (x',y') in (*Q)^2 (the enlargement of Q) that are not trivial and satisfy the enlarged equation (with equality replaced by infinitesimal closeness)


Now another theorem in NSA says that if x' is a hyper-rational then there is a "standard part" and a "nonstandard part" whose sum is x'.

Let x' = x_r + x_h where x_r is rational and x_h is a hyperrational infinitesimal.

We get (xr + xh)^3 + (yr + yh)^3 = 1 if x' ^3 + y' ^3 = 1.

Now I was working on this late last night so I need to check my work but I got yh = .5(1 + i * Sqrt(3)) yr.

If yr is rational this implies yh is NOT hyperRATIONAL, a contradiction implying that there are no hyperrationals (x,y) satisfying the equation. By transfer this implies that there are no rationals satisfying the equation.

Re: Nonstandard Algebraic Geometry

Posted: Mon Sep 05, 2016 9:25 pm
by Brian