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Full Rank Representation of Real Algebraic Sets and Applications

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Computer Algebra in Scientific Computing (CASC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10490))

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Abstract

We introduce the notion of the full rank representation of a real algebraic set, which represents it as the projection of a union of real algebraic manifolds \(V_{\mathbb {R}}(F_i)\) of \(\mathbb {R}^m\), \(m\ge n\), such that the rank of the Jacobian matrix of each \(F_i\) at any point of \(V_{\mathbb {R}}(F_i)\) is the same as the number of polynomials in \(F_i\).

By introducing an auxiliary variable, we show that a squarefree regular chain T can be transformed to a new regular chain C having various nice properties, such as the Jacobian matrix of C attains full rank at any point of \(V_{\mathbb {R}}(C)\). Based on a symbolic triangular decomposition approach and a numerical critical point technique, we present a hybrid algorithm to compute a full rank representation.

As an application, we show that such a representation allows to better visualize plane and space curves with singularities. Effectiveness of this approach is also demonstrated by computing witness points of polynomial systems having rank-deficient Jacobian matrices.

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References

  1. Alvandi, P., Chen, C., Hashemi, A., Maza, M.M.: Regular chains under linear changes of coordinates and applications. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2015. LNCS, vol. 9301, pp. 30–44. Springer, Cham (2015). doi:10.1007/978-3-319-24021-3_3

    Chapter  Google Scholar 

  2. Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28(1–2), 105–124 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brake, D., Hauenstein, J., Liddell, A.: Validating the completeness of the real solution set of a system of polynomial equations. ISSAC 2016, 143–150 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Caire, L.: Plane curves as projections of non singular space curves. Manuscripta Math. 67(1), 433–450 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, C., Davenport, J., May, J., Moreno Maza, M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. J. Symb. Comput. 49, 3–26 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, C., Wu, W.: A numerical method for computing border curves of bi-parametric real polynomial systems and applications. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 156–171. Springer, Cham (2016). doi:10.1007/978-3-319-45641-6_11

    Chapter  Google Scholar 

  8. Daouda, D., Mourrain, B., Ruatta, O.: On the computation of the topology of a non-reduced implicit space curve. ISSAC 2008, 47–54 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, Heidelberg (2013). doi:10.1007/978-1-4612-5350-1

    MATH  Google Scholar 

  10. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, Heidelberg (1997). doi:10.1007/978-1-4757-3849-0

    MATH  Google Scholar 

  11. Hauenstein, J.D.: Numerically computing real points on algebraic sets. Acta Appl. Math. 125(1), 105–119 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hauenstein, J., Sommese, A.: What is numerical algebraic geometry. J. Symb. Comp. 79, 499–507 (2017). Part 3

    Article  MathSciNet  MATH  Google Scholar 

  13. Hong, H., El Din, M.S.: Variant quantifier elimination. J. Symb. Comp. 47(7), 883–901 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hong, H.: An efficient method for analyzing the topology of plane real algebraic curves. Math. Comput. Simul. 42(4), 571–582 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Imbach, R., Moroz, G., Pouget, M.: Numeric and certified isolation of the singularities of the projection of a smooth space curve. MACIS 2015, 78–92 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Jin, K., Cheng, J.: Isotopic epsilon-meshing of real algebraic space curves. SNC 2014, 118–127 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Jin, K., Cheng, J.-S., Gao, X.-S.: On the topology and visualization of plane algebraic curves. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2015. LNCS, vol. 9301, pp. 245–259. Springer, Cham (2015). doi:10.1007/978-3-319-24021-3_19

    Chapter  Google Scholar 

  18. Labs, O.: A list of challenges for real algebraic plane curve visualization software. In: Emiris, I.Z., Sottile, F., Theobald, T. (eds.) Nonlinear Computational Geometry, pp. 137–164. Springer, New York (2010)

    Google Scholar 

  19. Lasserre, J., Laurent, M., Rostalski, P.: Semidefinite characterization and computation of zero-dimensional real radical ideals. Found. Comput. Math. 8(5), 607–647 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lecerf, G.: Quadratic newton iteration for systems with multiplicity. Found. Comput. Math. 2(3), 247–293 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, Heidelberg (2003). doi:10.1007/978-1-4419-9982-5

    Google Scholar 

  22. Leykin, A.: Numerical primary decomposition. ISSAC 2008, 165–172 (2008)

    Article  MathSciNet  Google Scholar 

  23. Leykin, A., Verschelde, J., Zhao, A.: Newton’s method with deflation for isolated singularities of polynomial systems. TCS 359(1), 111–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, T.Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numerica 6, 399–436 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ma, Y., Wang, C., Zhi, L.: A certificate for semidefinite relaxations in computing positive-dimensional real radical ideals. J. Symb. Comput. 72, 1–20 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mantzaflaris, A., Mourrain, B.: Deflation and certified isolation of singular zeros of polynomial systems. ISSAC 2011, 249–256 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Morrow, J.A., Kodaira, K.: Complex Manifolds, vol. 355. American Mathematical Society, Providence (1971)

    MATH  Google Scholar 

  28. Ojika, T., Watanabe, S., Mitsui, T.: Deflation algorithm for the multiple roots of a system of nonlinear equations. J. Math. Anal. Appl. 96(2), 463–479 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  29. Parrilo, P.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2), 293–320 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rouillier, F., Roy, M.F., El Din, M.S.: Finding at least one point in each connected component of a real algebraic set defined by a single equation. J. Complex. 16(4), 716–750 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. El Din, M.S., Schost, É.: Properness defects of projections and computation of at least one point in each connected component of a real algebraic set. Discrete Comput. Geom. 32(3), 417–430 (2004)

    MathSciNet  MATH  Google Scholar 

  32. El Din, M.S., Spaenlehauer, P.: Critical point computations on smooth varieties: degree and complexity bounds. In: ISSAC 2016, pp. 183–190 (2016)

    Google Scholar 

  33. Sommese, A., Verschelde, J., Wampler, C.: Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6), 2022–2046 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sommese, A., Wampler, C.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific Press, Singapore (2005)

    Book  MATH  Google Scholar 

  35. Wu, W., Reid, G.: Finding points on real solution components and applications to differential polynomial systems. ISSAC 2013, 339–346 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Hoon Hong and anonymous reviewers for their helpful comments. This work is partially supported by the projects NSFC (11471307, 11671377, 61572024), cstc2015jcyjys40001, and the Key Research Program of Frontier Sciences of CAS (QYZDB-SSW-SYS026).

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Authors and Affiliations

  1. Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, China

    Changbo Chen, Wenyuan Wu & Yong Feng

  2. University of Chinese Academy of Sciences, Beijing, China

    Changbo Chen, Wenyuan Wu & Yong Feng

Authors
  1. Changbo Chen
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  2. Wenyuan Wu
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  3. Yong Feng
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Correspondence to Wenyuan Wu .

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Editors and Affiliations

  1. Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Chen, C., Wu, W., Feng, Y. (2017). Full Rank Representation of Real Algebraic Sets and Applications. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_5

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_5

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