Uniqueness of Steiner laws on cubic curves

R. Padmanabhan and W. McCune

Abstract: In this paper we use the Cayley-Bacharach 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 Riemann-Roch theorem, the $\cal P$-function of Weierstrass, the rigidity lemma, Euler numbers, Lefschetz fixed-point theorem, and so on. However, our equational proofs employ automated reasoning by transforming the Cayley-Bacharach 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, Cayley-Bacharach theorem, $n$-ary composition laws, Steiner laws, geometric constructions, uniqueness theorems, automated reasoning, Otter, inference rules

Classification (MSC2000): 14N05, 20N15, 51M15, 68T15

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