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Cayley Factorization and the Area Principle

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Abstract

In Sturmfels and Whiteley (J Symb Comput 11(5):439–453, 1991), it is proven that any multihomogenous bracket polynomial with integer coefficients can be interpreted by a ruler construction when introducing simple non-degeneracy conditions. This problem is called (generalized) Cayley factorization. This allows for geometrically interpreting many projective invariant properties. We reprove the above statement in rank 3 giving a better bound on the size of the non-degeneracy conditions. The constant factor in the bound is essentially reduced from 105 to 9. The algorithm described is concise enough to be implemented on a computer. With this algorithm, interpreting a common condition for six points to lie on a common conic is very close to Pascal’s construction (see Apel, in: The geometry of brackets and the area principle, Dissertation, TU München, 2014).

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Notes

  1. For evaluating the formula given in the case \(m\ge 2\), use the second version of (1) within the equivalent formula .

References

  1. Apel, S.: Cayley factorization and the area principle. Papers presented at ADG 2012: The 9th International Workshop on Automated Deduction in Geometry, University of Edinburgh, 17–19 Sept 2012, pp. 33–41 (2012)

  2. Apel, S.: The geometry of brackets and the area principle. Dissertation, Fakultät für Mathematik, Technische Universität München (2014). http://mediatum.ub.tum.de/node?id=1175107

  3. Barnabei, M., Brini, A., Rota, G.-C.: On the exterior calculus of invariant theory. J. Algebra 96(1), 120–160 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berge, C.: Two theorems in graph theory. Proc. Natl Acad. Sci. USA 43(9), 842–844 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  5. Browne, J.: Grassmann Algebra: Foundations: Exploring Extended Vector Algebra with Mathematica. CreateSpace Independent Publishing Platform (2012)

  6. Carrell, J.B., Dieudonné, J.A.: Invariant Theory, Old and New. Academic Press, New York (1971)

    MATH  Google Scholar 

  7. Chan, W., Rota, G.-C., Stein, J.A.: The power of positive thinking. In: White, N.L. (ed.) Invariant Methods in Discrete and Computational Geometry, pp. 1–36. Springer, Dordrecht (1995)

    Chapter  Google Scholar 

  8. Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry Theorems, vol. 6. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  9. Crapo, H.: Invariant-theoretic methods in scene analysis and structural mechanics. J. Symb. Comput. 11(5), 523–548 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Crapo, H.: Projective configurations: their invariants and homology. Proc. R. Ir. Acad. Sect. A 95A(Supplement), 35–58 (1995)

    MathSciNet  Google Scholar 

  11. Crapo, H., Rota, G.-C.: On the Foundations of Combinatorial Theory: Combinatorial Geometries. MIT Press, Cambridge (1970)

    MATH  Google Scholar 

  12. Crapo, H., Rota, G.-C.: The resolving bracket. In: White, N.L. (ed.) Invariant Methods in Discrete and Computational Geometry, pp. 197–222. Springer, Dordrecht (1995)

    Chapter  Google Scholar 

  13. Crapo, H., Whiteley, W.: Statics of frameworks and motions of panel structures: a projective geometric introduction. Struct. Topol. 6, 43–82 (1982)

    MathSciNet  Google Scholar 

  14. Crelle, A.L.: Ueber einige Eigenschaften des ebenen geradlinigen Dreiecks rücksichtlich dreier durch die Winkel-Spitzen gezogenen geraden Linien, Berlin (1816)

  15. Diestel, R.: Graph Theory, 2nd edn. Springer, New York (2000)

    Google Scholar 

  16. Doubilet, P., Rota, G.-C., Stein, J.: On the foundations of combinatorial theory IX. Combinatorial methods in invariant theory. Stud. Appl. Math. 53(3), 185–216 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  17. Grosshans, F.D., Rota, G.-C., Stein, J.A.: Invariant Theory and Superalgebras. American Mathematical Society, Providence, RI (1987)

    Book  Google Scholar 

  18. Grünbaum, B., Shephard, G.C.: Ceva, Menelaus, and the area principle. Math. Mag. 68(4), 254–268 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Grünbaum, B., Shephard, G.C.: Some new transversality properties. Geom. Dedicata 71(2), 179–208 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hall, P.: On representatives of subsets. J. Lond. Math. Soc. 10(1), 26–30 (1935)

    Google Scholar 

  21. Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry, vol. 1. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  22. Huang, R.Q., Rota, G.-C., Stein, J.A.: Supersymmetric bracket algebra and invariant theory. In: Piacentini Cattaneo, G.M., Strickland, E. (eds.) Topics in Computational Algebra, pp. 193–246. Springer, Dordrecht (1990)

    Chapter  Google Scholar 

  23. Kraft, H., Procesi, C.: Classical Invariant Theory, a Primer. Lecture Notes, 125 pp., http://jones.math.unibas.ch/~kraft/Papers/KP-Primer.pdf (1996)

  24. Li, H.: Invariant Algebras and Geometric Reasoning. World Scientific, Singapore (2008)

    Book  MATH  Google Scholar 

  25. Li, H., Wu, Y.: Automated short proof generation for projective geometric theorems with Cayley and bracket algebras: I. Incidence geometry. J. Symb. Comput. 36(5), 717–762 (2003)

    Article  MATH  Google Scholar 

  26. Li, H., Zhao, L., Chen, Y.: Polyhedral scene analysis combining parametric propagation with calotte analysis. In: Li, H., Olver, P.J., Sommer, G. (eds.) Computer Algebra and Geometric Algebra with Applications, pp. 383–402. Springer, Berlin (2005)

    Chapter  Google Scholar 

  27. Möbius, A.F.: Der barycentrische Calcul. Johann Ambrosius Barth, Berlin (1827)

    Google Scholar 

  28. Richter-Gebert, J.: Perspectives on Projective Geometry. Springer, New York (2011)

    Book  MATH  Google Scholar 

  29. Sturmfels, B.: Algorithms in Invariant Theory. Springer, New York (1993)

    Book  MATH  Google Scholar 

  30. Sturmfels, B., Whiteley, W.: On the synthetic factorization of projectively invariant polynomials. J. Symb. Comput. 11(5), 439–453 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tay, T.-S.: On the Cayley factorization of calotte conditions. Discrete Comput. Geom. 11(1), 97–109 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  32. Weyl, H.: The Classical Groups: Their Invariants and Representations, vol. 1. Princeton University Press, Princeton (1997)

    MATH  Google Scholar 

  33. White, N.L.: The bracket ring of a combinatorial geometry. I. Trans. Am. Math. Soc. 202, 79–95 (1975)

    Article  MATH  Google Scholar 

  34. White, N.L.: Cayley factorization and a straightening algorithm. In: Piacentini Cattaneo, G.M., Strickland, E. (eds.) Topics in Computational Algebra, pp. 163–184. Springer, Dordrecht (1990)

    Chapter  Google Scholar 

  35. White, N.L.: Multilinear Cayley factorization. J. Symb. Comput. 11(5), 421–438 (1991)

    Article  MATH  Google Scholar 

  36. White, N.L.: Grassmann–Cayley algebra and robotics. J. Intell. Robot. Syst. 11(1–2), 91–107 (1994)

    Article  MATH  Google Scholar 

  37. White, N.L.: A tutorial on Grassmann–Cayley algebra. In: White, N.L. (ed.) Invariant Methods in Discrete and Computational Geometry, pp. 93–106. Springer, Dordrecht (1995)

    Chapter  Google Scholar 

  38. White, N.L.: Grassmann–Cayley algebra and robotics applications. In: Bayro-Corrochano, E. (ed.) Handbook of Geometric Computing, pp. 629–656. Springer, Berlin (2005)

    Chapter  Google Scholar 

  39. White, N.L., McMillan, T.: Cayley factorization. In: Gianni, P. (ed.) Symbolic and Algebraic Computation, pp. 521–533. Springer, Berlin (1989)

    Chapter  Google Scholar 

  40. White, N.L., Whiteley, W.: The algebraic geometry of stresses in frameworks. SIAM J. Algebraic Discrete Methods 4(4), 481–511 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  41. Whiteley, W.: Invariant computations for analytic projective geometry. J. Symb. Comput. 11(5), 549–578 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  42. Whiteley, W.J.: Logic and invariant theory. Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology (1971)

  43. Wolfram Research, Inc.: Mathematica Edition: Version 9.0 (2012)

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  1. Zentrum Mathematik, Lehrstuhl für Geometrie und Visualisierung (M10), Technische Universität München, Boltzmannstr. 3, 85748, Garching bei München, Germany

    Susanne Apel

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Apel, S. Cayley Factorization and the Area Principle. Discrete Comput Geom 55, 203–227 (2016). https://doi.org/10.1007/s00454-015-9738-2

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