Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 47, No. 2, pp. 543557 (2006) 

Uniqueness of Steiner laws on cubic curvesR. Padmanabhan and W. McCuneDepartment of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada; email: padman@cc.umanitoba.ca; Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 604394844, U.S.A. email: mccune@mcs.anl.govAbstract: In this paper we use the CayleyBacharach theorem of classical algebraic geometry to construct several universal algebras on algebraic curves using divisors and complete intersection cycles and study the equational identities valid for these synthetic constructions. These results are not necessarily new; in fact, all of them may be ``easily'' provable by resorting to such powerful tools as the RiemannRoch theorem, the $\cal P$function of Weierstrass, the rigidity lemma, Euler numbers, Lefschetz fixedpoint theorem, and so on. However, our equational proofs employ automated reasoning by transforming the CayleyBacharach theorem into a formal implication. Besides being elementary, this approach provides new examples for model theorists and computer scientists designing theorem provers and gives new insights and interpretations for these various geometric constructions. Keywords: Cubic curves, CayleyBacharach theorem, $n$ary composition laws, Steiner laws, geometric constructions, uniqueness theorems, automated reasoning, Otter, inference rules Classification (MSC2000): 14N05, 20N15, 51M15, 68T15 Full text of the article:
Electronic version published on: 19 Jan 2007. This page was last modified: 5 Nov 2009.
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