An equational characterization of the conic construction of cubic curves

An n-ary Steiner law f(x{sub 1},x{sub 2},{hor_ellipsis},x{sub n}) on a projective curve {Gamma} over an algebraically closed field k is a totally symmetric n-ary morphism f from {Gamma}{sup n} to {Gamma} satisfying the universal identity f(x{sub 1},x{sub 2},{hor_ellipsis},x{sub n-1}, f(x{sub 1},x{sub 2},{hor_ellipsis},x{sub n})) = x{sub n}. An element e in {Gamma} is called an idempotent for f if f(e,e,{hor_ellipsis},e) = e. The binary morphism x * y of the classical chord-tangent construction on a nonsingular cubic curve is an example of a binary Steiner law on the curve, and the idempotents of * are precisely the inflection points of the curve. In this paper, the authors prove that if f and g are two 5-ary Steiner laws on an elliptic curve {Gamma} sharing a common idempotent, then f = g. They use a new rule of inference rule =(gL){implies}, extracted from a powerful local-to-global principal in algebraic geometry. This rule is implemented in the theorem-proving program OTTER. Then they use OTTER to automatically prove the uniqueness of the 5-ary Steiner law on an elliptic curve. Very much like the binary case, this theorem provides an algebraic characterization of a geometric construction process involving conics and cubics. The well-known theorem of the uniqueness of the group law on such a curve is shown to be a consequence of this result.