An equational characterization of the conic construction of cubic curves

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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 ... continued below

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11 p.

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McCune, W. & Padmanabhan, R. May 17, 1995.

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  • McCune, W. Argonne National Lab., IL (United States)
  • Padmanabhan, R. Univ. of Manitoba, Winnipeg (Canada). Dept. of Mathematics

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Description

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.

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11 p.

Notes

OSTI as DE97008538

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  • Other Information: PBD: 17 May 1995

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  • Other: DE97008538
  • Report No.: ANL/ER/PP--86642
  • Grant Number: W-31109-ENG-38
  • DOI: 10.2172/516005 | External Link
  • Office of Scientific & Technical Information Report Number: 516005
  • Archival Resource Key: ark:/67531/metadc697791

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  • May 17, 1995

Added to The UNT Digital Library

  • Aug. 14, 2015, 8:43 a.m.

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  • Dec. 16, 2015, 12:24 p.m.

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McCune, W. & Padmanabhan, R. An equational characterization of the conic construction of cubic curves, report, May 17, 1995; Illinois. (digital.library.unt.edu/ark:/67531/metadc697791/: accessed September 21, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.