Algebraic independence of the universal Weierstrass invariants

The invariants c₄ and c₆ of the universal five-coefficient Weierstrass equation are algebraically independent over ℤ. Thus the subalgebra they generate is a polynomial algebra on two generators. This is the independence input for determining the relations among c₄, c₆ and the discriminant, whose known relation is c₄³ - c₆² = 1728Δ.

The argument specializes the universal equation to the short equation with prescribed invariants over the rational function field ℚ(C₄, C₆). It uses the existing WeierstrassCurve.ofCInvariants and its invariant formulas; no integrality assertion about that short equation is needed. The resulting independence is integral, even though its specialization is rational.

theorem WeierstrassCurve.Universal.algebraicIndependent_c₄_c₆ : AlgebraicIndependent ℤ ![c₄ curve, c₆ curve]

The two c-invariants of the universal Weierstrass equation satisfy no nonzero polynomial relation with integer coefficients. In particular, the relation involving the discriminant does not impose a relation on c₄ and c₆ alone.