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...Zvi rosen algebraic geometry notes richard borcherds thursday august course website http math berkeley edu reb a text hartshorne example find solutions x y z to solution assume coprime and odd so are both squares set r s with positive numbers rs then for taking gives us geometric solve in rationals where therefore we looking rational points on the circle finding real is easy just take sin cos however this doesn t help very much instead point at look lines from that intersect tangent do these if line intersects it almost correspond means map birational isomorphism except o codimension smooth manifolds maps trivial any manifold of dimension n can be cut along dimensional sub union rns because group rotations pythagorean triples product formula works send commutative ring integers one wecan draw graph insert here has singularity origin will slope given determine solving give cubic equation two roots third root tx taste singularities which generally want get rid accom plish through resolut...