Department of Mathematics
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Last week Dr. Peter Penner proved that the identities satisfied by linear forms over reals are a consequence of the inference rule (gL) and the median law. In this talk, we show the geometric relevance of these two algebraic principles. Apart from being an hyper-identity for all abelian group terms, the median law is also valid for many classical geometric constructions (e.g. on conics and cubics) while the rule (gL) may be construed as an avatar of an intersection theorem in algebraic geometry (see e.g. [1]). In fact, we will derive the validity of the binary median law for cubic curves from the Cayley-Bacharach theorem. The associativity of the group law on elliptic curves as well as the classical Pappus-Pascal theorems for conics are easy consequences of the median law (see e.g. [3], page 407 and [4], page 69).

References

1. D. Eisenbud et al., Cayley-Bacharach Theorems, Bulletin AMS 33 (1966) 295-324.
2. T. Evans, Properties of algebras almost equivalent to identities. J. London Math. Soc. 37 (1962) 53--59.
3. R. Harshorne, Algebraic Geometry, Springer, 1977.
4. A. Knapp, Elliptic Curves, Princeton, 1993.
5. R. Padmanabhan and P. Penner, An implication basis for linear forms, Algebra Universalis, (2006).