Equational characterization of the conic construction of cubic curves

摘要

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.