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Heuristic Algorithms for Recognition of Some Cubic Hypersurfaces

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Abstract

In this paper, we propose some heuristic probabilistic polynomial time algorithms with one-sided error for recognition of cubic hypersurfaces the singular loci of which do not contain any linear subspace of sufficiently large dimension. These algorithms are easy to implement in computer algebra systems. The algorithms are based on checking the condition that the Hessian determinant of a cubic form does not vanish identically or does not determine any cone in the projective space. In turn, the properties of the Hessian can be verified with one-sided-error probabilistic algorithms based on the Schwartz–Zippel lemma.

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REFERENCES

  1. Alaev, P.E. and Selivanov, V.L., Fields of algebraic numbers computable in polynomial time. I, Algebra Logic, 2020, vol. 58, pp. 447–469. https://doi.org/10.1007/s10469-020-09565-0

    Article  MATH  Google Scholar 

  2. Malaschonok, G.I., MathPartner computer algebra, Program. Comput. Software, 2017, vol. 43, no. 2, pp. 112–118. https://doi.org/10.1134/S0361768817020086

    Article  MathSciNet  Google Scholar 

  3. Neumann, E. and Pauly, A., A topological view on algebraic computation models, J. Complexity, 2018, vol. 44, pp. 1–22. https://doi.org/10.1016/j.jco.2017.08.003

    Article  MathSciNet  MATH  Google Scholar 

  4. Abramov, S.A., Lektsii o slozhnosti algoritmov (Lectures on Complexity of Algorithms), Moscow: Mosk. Tsentr Nepreryvnogo Mat. Obraz., 2012.

  5. Rybalov, A.N., Generic amplification of recursively enumerable sets, Algebra Logic, 2018, vol. 57, no. 4, pp. 289–294. https://doi.org/10.1007/s10469-018-9500-y

    Article  MathSciNet  MATH  Google Scholar 

  6. Rybalov, A.N., On generic NP-completeness of the problem of Boolean circuits satisfiability, Prikl. Diskretnaya Mat., 2020, no. 47, pp. 101–107. https://doi.org/10.17223/20710410/47/8

  7. Seliverstov, A.V., On binary solutions of systems to equations, Prikl. Diskretnaya Mat., 2019, no. 45, pp. 26–32. https://doi.org/10.17223/20710410/45/3

  8. Eder, C. and Faugére, J.-C., A survey on signature-based algorithms for computing Gröbner bases, J. Symbolic Comput., 2017, vol. 80, no. 3, pp. 719–784. https://doi.org/10.1016/j.jsc.2016.07.031

    Article  MathSciNet  MATH  Google Scholar 

  9. Yanovich, D.A., Computation of involutive and Gröbner bases using the tableau representation of polynomials, Program. Comput. Software, 2020, vol. 46, no. 2, pp. 162–166. https://doi.org/10.1134/S0361768820020115

    Article  Google Scholar 

  10. Brzostowski, S., Krasiński, T., and Walewska, J., Arnold’s problem on monotonicity of the Newton number for surface singularities, J. Math. Soc. Jpn., 2019, vol. 71, no. 4, pp. 1257–1268. https://doi.org/10.2969/jmsj/78557855

    Article  MathSciNet  MATH  Google Scholar 

  11. Bryuno, A.D., The asymptotic behavior of solutions of nonlinear systems of differential equations, Sov. Math. Dokl., 1962, vol. 3, pp. 464–467.

    MATH  Google Scholar 

  12. Bryuno, A.D., Algorithms for solving an algebraic equation, Program. Comput. Software, 2018, vol. 44, no. 6, pp. 533–545. https://doi.org/10.1134/S0361768819100013

    Article  MathSciNet  Google Scholar 

  13. Bryuno, A.D., On the parametrization of an algebraic curve, Math. Notes, 2019, vol. 106, no. 6, pp. 885–893. https://doi.org/10.1134/S0001434619110233

    Article  MathSciNet  MATH  Google Scholar 

  14. Antipova, I.A., Mikhalkin, E.N., and Tsikh, A.K., Singular points of complex algebraic hypersurfaces, J. Sib. Fed. Univ. Math. Phys., 2018, vol. 11, no. 6, pp. 670–679. https://doi.org/10.17516/1997-1397-2018-11-6-670-679

    Article  MathSciNet  Google Scholar 

  15. Gel’fand, I.M., Zelevinskii, A.V., and Kapranov, M.M., Discriminants of polynomials in many variables, Funct. Anal. Appl., 1990, vol. 24, no. 1, pp. 1–4. https://doi.org/10.1007/BF01077912

    Article  MathSciNet  Google Scholar 

  16. Seliverstov, A.V., On tangent lines to affine hypersurfaces, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 2017, vol. 27, no. 2, pp. 248–256. https://doi.org/10.20537/vm170208

    Article  MathSciNet  MATH  Google Scholar 

  17. Rubanov, L.I. and Seliverstov, A.V., Projective-invariant description of a meandering river, J. Commun. Technol. Electron., 2017, vol. 62, no. 6, pp. 663–668. https://doi.org/10.1134/S1064226917060201

    Article  Google Scholar 

  18. van Hoeij, M., An algorithm for computing the Weierstrass normal form, Proc. Int. Symp. Symbolic and Algebraic Computation (ISSAC), Levelt, A.H.M., Ed., New York: ACM Press, 1995, pp. 90–95.

  19. Slyadnev, S.E. and Turlapov, V.E., Simplification of CAD models by automatic recognition and suppression of blend chains, Program. Comput. Software, 2020, vol. 46, no. 3, pp. 233–243. https://doi.org/10.1134/S0361768820030081

    Article  MathSciNet  Google Scholar 

  20. Segre, B., A note on arithmetical properties of cubic surfaces, J. London Math. Soc., 1943, vol. 18, pp. 24–31.

    Article  MathSciNet  Google Scholar 

  21. Kollár, J., Unirationality of cubic hypersurfaces, J. Inst. Math. Jussieu, 2002, vol. 1, no. 3, pp. 467–476. https://doi.org/10.1017/S1474748002000117

    Article  MathSciNet  MATH  Google Scholar 

  22. Polo-Blanco, I. and Top, J., A remark on parameterizing nonsingular cubic surfaces, Comput.-Aided Geom. Des., 2009, vol. 26, no. 8, pp. 842–849. https://doi.org/10.1016/j.cagd.2009.06.001

    Article  MathSciNet  MATH  Google Scholar 

  23. González-Sánchez, J. and Polo-Blanco, I., Construction algorithms for rational cubic surfaces, J. Symbolic Comput., 2017, vol. 79, pp. 309–326. https://doi.org/10.1016/j.jsc.2016.02.010

    Article  MathSciNet  MATH  Google Scholar 

  24. Pan, V.Ya., Fast matrix multiplication and its algebraic neighbourhood, Sb.: Math., 2017, vol. 208, no. 11, pp. 1661–1704. https://doi.org/10.1070/SM8833

    Article  MathSciNet  MATH  Google Scholar 

  25. Malaschonok, G., Recursive matrix algorithms, distributed dynamic control, scaling, stability, Proc. Workshop Computer Science and Information Technologies (CSIT), Yerevan, Armenia, 2019, pp. 112–115. https://doi.org/10.1109/CSITechnol.2019.8895255

  26. Schwartz, J.T., Fast probabilistic algorithms for verification of polynomial identities, J. ACM, 1980, vol. 27, no. 4, pp. 701–717. https://doi.org/10.1145/322217.322225

    Article  MathSciNet  MATH  Google Scholar 

  27. Gondim, R. and Russo, F., On cubic hypersurfaces with vanishing hessian, J. Pure Appl. Algebra, 2015, vol. 219, no. 4, pp. 779–806. https://doi.org/10.1016/j.jpaa.2014.04.030

    Article  MathSciNet  MATH  Google Scholar 

  28. Bibikov, P.V., Classification of ternary forms with zero hessian, Russ. Math., 2011, vol. 55, no. 9, pp. 83–85. https://doi.org/10.3103/S1066369X11090118

    Article  MathSciNet  MATH  Google Scholar 

  29. Seliverstov, A.V., Symmetric matrices whose entries are linear functions, Comput. Math. Math. Phys., 2020, vol. 60, pp. 102–108. https://doi.org/10.1134/S0965542520010121

    Article  MathSciNet  MATH  Google Scholar 

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ACKNOWLEDGMENTS

We are grateful to Mark Spivakovsky for his participation in discussion of this work, as well as to S.A. Abramov and A.B. Batkhin for their helpful remarks.

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  1. Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, 127051, Moscow, Russia

    A. V. Seliverstov

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Translated by Yu. Kornienko

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Seliverstov, A.V. Heuristic Algorithms for Recognition of Some Cubic Hypersurfaces. Program Comput Soft 47, 50–55 (2021). https://doi.org/10.1134/S0361768821010096

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