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Multivariate Resultants in Bernstein Basis

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Automated Deduction in Geometry (ADG 2008)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6301))

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Abstract

Macaulay and Dixon resultant formulations are proposed for parametrized multivariate polynomial systems represented in Bernstein basis. It is proved that the Macaulay resultant for a polynomial system in Bernstein basis vanishes for the total degree case if and only if the either the polynomial system has a common Bernstein-toric root, a common infinite root, or the leading forms of the polynomial system obtained by replacing every variable x i in the original polynomial system by \(\frac{y_i}{1+y_i}\) have a non-trivial common root. For the Dixon resultant formulation, the rank sub-matrix constructions for the original system and the transformed system are shown to be essentially equivalent. Known results about exactness of Dixon resultants of a sub-class of polynomial systems as discussed in Chtcherba and Kapur in Journal of Symbolic Computation (August, 2003) carry over to polynomial systems represented in the Bernstein basis. Furthermore, in certain cases, when the extraneous factor in a projection operator constructed from the Dixon resultant formulation is precisely known, such results also carry over to projection operators of polynomial systems in the Bernstein basis where extraneous factors are precisely known. Applications of these results in the context of geometry theorem proving, implicitization and intersection of surfaces with curves are discussed. While Macaulay matrices become large when polynomials in Bernstein bases are used for problems in these applications, Dixon matrices are roughly of the same size.

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References

  1. Amiraslani, A.: Dividing polynomials when you only know their values. In: Gonzalez-Vega, L., Recio, T. (eds.) Proceedings of Encuentros de Álgebra Computacional y Aplicaciones (EACA) 2004, pp. 5–10 (2004)

    Google Scholar 

  2. Barnett, S.: Polynomials and linear control systems. Monographs and Textbooks in Pure and Applied Mathematics, vol. 77. Marcel Dekker Inc., New York (1983)

    MATH  Google Scholar 

  3. Barnett, S.: Division of generalized polynomials using the comrade matrix. Linear Algebra Appl. 60, 159–175 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barnett, S.: Euclidean remainders for generalized polynomials. Linear Algebra Appl. 99, 111–122 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berchtold, J., Bowyer, A.: Robust arithmetic for multivariate bernstein-form polynomials. In: Computer-Aided Design, pp. 681–689 (2000)

    Google Scholar 

  6. Bini, D.A., Gemignani, L.: Bernstein-Bezoutian matrices. Theoret. Comput. Sci. 315(2-3), 319–333 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bini, D.A., Gemignani, L., Winkler, J.R.: Structured matrix methods for CAGD: an application to computing the resultant of polynomials in the Bernstein basis. Numer. Linear Algebra Appl. 12(8), 685–698 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brazier, M., Chcherba, A.: MatDetInterp. Symbolic matrix determinant interpolator, http://www.chtcherba.com/arthur/Projects/MatDetInterp/

  9. Busé, L., Elkadi, M., Mourrain, B.: Generalized resultants over unirational algebraic varieties. J. Symbolic Computation 29(4-5), 515–526 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Canny, J.: Generalised characteristic polynomials. J. Symbolic Computation 9, 241–250 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cheng, H., Labahn, G.: On computing polynomial GCDs in alternate bases. In: ISSAC 2006, pp. 47–54. ACM, New York (2006)

    Google Scholar 

  12. Chtcherba, A., Kapur, D.: Exact resultants for corner-cut unmixed multivariate polynomial systems using the Dixon formulation. J. Symbolic Computation 36(3-4), 289–315 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chtcherba, A.D., Kapur, D., Minimair, M.: Cayley-dixon resultant matrices of multi-univariate composed polynomials. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 125–137. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  14. Chtcherba, A.D.: A new Sylvester-type Resultant Method based on the Dixon-Bézout Formulation. PhD dissertation, University of New Mexico, Department of Computer Science (August 2003)

    Google Scholar 

  15. Chtcherba, A.D., Kapur, D.: Conditions for determinantal formula for resultant of a polynomial system. In: ISSAC 2006: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, Genoa, Italy, pp. 55–62. ACM, New York (2006), doi:10.1145/1145768.1145784

    Google Scholar 

  16. Corless, R.: Generalized companion matrices in the lagrange basis. In: Gonzalez-Vega, L., Recio, T. (eds.) Proceedings of Encuentros de Álgebra Computacional y Aplicaciones (EACA) 2004, pp. 317–322 (2004)

    Google Scholar 

  17. Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, Heidelberg (1998)

    Book  MATH  Google Scholar 

  18. D’Andrea, C.: Macaulay style formulas for sparse resultants. Trans. Amer. Math. Soc. 354(7), 2595–2629 (electronic) (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Devore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Heidelberg (1993)

    Book  MATH  Google Scholar 

  20. Diaz-Toca, G.M., Gonzalez-Vega, L.: Barnett’s theorems about the greatest common divisor of several univariate polynomials through Bezout-like matrices. J. Symbolic Comput. 34(1), 59–81 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dixon, A.-L.: On a form of the elimination of two quantics. Proc. London Math. Soc. 6, 468–478 (1908)

    Article  MATH  Google Scholar 

  22. Farin, G.F.: Curves and Surfaces for CAGD: A practical guide, 5th edn. Morgan Kaufmann, San Francisco (1991)

    Google Scholar 

  23. Gemignani, L.: Manipulating polynomials in generalized form. Tech. Rep. TR-96-14, Università di Pisa, Departmento di Informatica, Corso Italia 40, 56125 Pisa, Italy (December 1996)

    Google Scholar 

  24. Heymann, W.: Problem der Winkelhalbierenden. Ztschr. f. Math. und Phys. 35 (1890)

    Google Scholar 

  25. Kapur, D., Saxena, T.: Sparsity considerations in Dixon resultants. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, PA, pp. 184–191. ACM, New York (1996)

    Google Scholar 

  26. Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning using the Dixon resultants. In: ACM ISSAC 1994, Oxford, England, pp. 99–107 (July 1994)

    Google Scholar 

  27. Lewis, R.: Comparing acceleration techniques for the Dixon and Macaulay resultants. Mathematics and Computers in Simulation (2008) (accepted)

    Google Scholar 

  28. Macaulay, F.S.: The algebraic theory of modular systems. Cambridge Mathematical Library (1916)

    Google Scholar 

  29. Mani, V., Hartwig, R.E.: Generalized polynomial bases and the Bezoutian. Linear Algebra Appl. 251, 293–320 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Manocha, D., Krishnan, S.: Algebraic pruning: A fast technique for curve and surface intersection. Computer-Aided Geometric Design 20, 1–23 (1997)

    MathSciNet  MATH  Google Scholar 

  31. Maroulas, J., Barnett, S.: Greatest common divisor of generalized polynomial and polynomial matrices. Linear Algebra Appl. 22, 195–210 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Minimair, M.: MR, macaulay resultant package for Maple (April 2003), http://minimair.org/MR.mpl

  33. Minimair, M.: Basis-independent polynomial division algorithm applied to division in lagrange and bernstein basis (CD-ROM). In: Kapur, D. (ed.) Proceedings of Asian Symposium on Computer Mathematics (ASCM). National University of Singapore (2007)

    Google Scholar 

  34. Minimair, M.: DR, Maple package for computing Dixon projection operators (resultants) (2007), http://minimair.org/dr

  35. Tsai, Y.-F., Farouki, R.T.: Algorithm 812: BPOLY: An object-oriented library of numerical algorithms for polynomials in Bernstein form. ACM Transactions on Mathematical Software 27(2), 267–296 (2001)

    Article  MATH  Google Scholar 

  36. Winkler, J.R.: A resultant matrix for scaled Bernstein polynomials. Linear Algebra Appl. 319(1-3), 179–191 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Winkler, J.R.: Computational experiments with resultants for scaled Bernstein polynomials. In: Mathematical Methods for Curves and Surfaces, Oslo, pp. 535–544. Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN (2001)

    Google Scholar 

  38. Winkler, J.R.: Properties of the companion matrix resultant for Bernstein polynomials. In: Uncertainty in Geometric Computations. Kluwer Internat. Ser. Engrg. Comput. Sci., vol. 704, pp. 185–198. Kluwer Acad. Publ., Boston (2002)

    Chapter  Google Scholar 

  39. Winkler, J.R.: A companion matrix resultant for Bernstein polynomials. Linear Algebra Appl. 362, 153–175 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Winkler, J.R.: Numerical and algebraic properties of Bernstein basis resultant matrices. In: Computational Methods for Algebraic Spline Surfaces, pp. 107–118. Springer, Berlin (2005)

    Chapter  Google Scholar 

  41. Winkler, J.R., Goldman, R.N.: The Sylvester resultant matrix for Bernstein polynomials. In: Curve and Surface Design, Saint-Malo. Mod. Methods Math., pp. 407–416. Nashboro Press, Brentwood (2003)

    Google Scholar 

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Kapur, D., Minimair, M. (2011). Multivariate Resultants in Bernstein Basis. In: Sturm, T., Zengler, C. (eds) Automated Deduction in Geometry. ADG 2008. Lecture Notes in Computer Science(), vol 6301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21046-4_4

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  • DOI: https://doi.org/10.1007/978-3-642-21046-4_4

  • Publisher Name: Springer, Berlin, Heidelberg

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  • Online ISBN: 978-3-642-21046-4

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