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Hilbert Problem 15 and Ritt-Wu Method (I)

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Abstract

Hilbert problem 15 requires to understand Schubert’s book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near a 3-multiple straight line, which was obtained by the so called main trunk numbers, while for these numbers, Schubert said that he obtained them by experiences. So essentially Schubert even did not give any hint for the proof of this theorem. In this paper, by using the concept of generic point in the framework of Van der Waerden and Weil on algebraic geometry, and realizing Ritt-Wu method on computer, the authors prove that this theorem of Schubert is completely right.

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References

  1. Schubert H, Kalkül der abzählenden Geometrie, Springer-Verlag, 1979.

    Book  MATH  Google Scholar 

  2. Van der Waerden, Einfuehrung in die algebraische Geometrie, Verlag von Julius Springer, 1973, A Chinese translation has been published by Science Press in China in 2008.

  3. Weil A, Foundation of Algebraic Geometry, published by American Mathematical Society, 1946 and 1962.

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  4. Wang D K, Zero decomposition algorithms for system of polynomial equations, Proc. of the 4th Asian Symposium, Computer Mathematics, 2000, 67–70.

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  5. Chou S C, Schelter W F, and Yang J G, An algorithm for constructing gröbner bases from characteristic sets and its application to geometry, Algorithmica, 1990, 5: 147–154.

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang D M, Irreducible decomposition of algebraic varieties via characteristic set and Gröbner bases, Computer Aided Geometric Design, 1992, 9(6): 471–484.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thanks professor Dingkang WANG for many helps deeply

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Banghe Li

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  1. Banghe Li
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Correspondence to Banghe Li.

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Li, B. Hilbert Problem 15 and Ritt-Wu Method (I). J Syst Sci Complex 32, 47–61 (2019). https://doi.org/10.1007/s11424-019-8344-4

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  • DOI: https://doi.org/10.1007/s11424-019-8344-4

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