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## Archive of Issues

Russia Moscow
Year
2017
Volume
27
Issue
2
Pages
248-256
 Section Mathematics Title On tangent lines to affine hypersurfaces Author(-s) Seliverstov A.V.a Affiliations Institute for Information Transmission Problems, Russian Academy of Sciencesa Abstract The article focuses on methods to look for singular points of an affine hypersurface or to confirm the smoothness of the hypersurface. Our approach is based on the description of tangent lines to the hypersurface. The existence of at least one singular point imposes a restriction on the algebraic equation that determines the set of tangent lines passing through the selected point of the space. This equation is based on the formula for the discriminant of a univariate polynomial. For an arbitrary fixed hypersurface degree, we have proposed a deterministic polynomial time algorithm for computing a basis for the subspace of the corresponding polynomials. If a linear combination of these polynomials does not vanish on the hypersurface, then the hypersurface is smooth. We state a sufficient smoothness condition, which is verifiable in polynomial time. There are smooth affine hypersurfaces for which the condition is satisfied. The set includes the graphs of cubic polynomials in many variables as well as other examples of cubic hypersurfaces. On the other hand, the condition is violated for some high-dimensional cubic hypersurfaces. This does not prevent the application of the method in low dimensions. Searching for singular points is also important for solving some problems of machine vision, including detection of a corner by means of the frame sequence with one camera on a moving vehicle. Keywords hypersurface, singular point, tangent line, polynomial, discriminant UDC 514.14 MSC 14J70, 14Q10 DOI 10.20537/vm170208 Received 30 January 2017 Language Russian Citation Seliverstov A.V. On tangent lines to affine hypersurfaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 248-256. References Eder C., Faugère J.-C. A survey on signature-based algorithms for computing Gröbner bases, J. Symbolic Comput., 2017, vol. 80, part 3, pp. 719-784. DOI: 10.1016/j.jsc.2016.07.031 Malaschonok G.I. New generation of symbolic computation systems, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2016, vol. 21, issue 6, pp. 2026-2041 (in Russian). DOI: 10.20310/1810-0198-2016-21-6-2026-2041 Kulikov V.R., Stepanenko V.A. On solutions and Waring's formulae for the system of $n$ algebraic equations for $n$ unknowns, St. Petersburg Math. J., 2015, vol. 26, pp. 839-848. DOI: 10.1090/spmj/1361 Herrero M.I., Jeronimo G., Sabia J. Affine solution sets of sparse polynomial systems, J. Symbolic Comput., 2013, vol. 51, pp. 34-54. DOI: 10.1016/j.jsc.2012.03.006 Seliverstov A.V. Cubic forms without monomials in two variables, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, issue 1, pp. 71-77 (in Russian). DOI: 10.20537/vm150108 Gel'fand I.M., Zelevinskii A.V., Kapranov M.M. Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Leningrad Math. J., 1991, vol. 2, issue 3, pp. 499-505. Cenk M., Hasan M.A. On the arithmetic complexity of Strassen-like matrix multiplications, J. Symbolic Comput., 2017, vol. 80, pp. 484-501. DOI: 10.1016/j.jsc.2016.07.004 Polo-Blanco I., Top J. A remark on parameterizing nonsingular cubic surfaces, Comput. Aided Geom. Design, 2009, vol. 26, issue 8, pp. 842-849. DOI: 10.1016/j.cagd.2009.06.001 Seliverstov A.V. On cubic hypersurfaces with involutions, International Conference Polynomial Computer Algebra'2016, Russian Academy of Sciences, St. Petersburg Department of Steklov Mathematical Institute, Euler International Mathematical Institute, St. Petersburg, 2016, pp. 74-77. http://elibrary.ru/item.asp?id=26437524 Golubyatnikov V.P. On unique recoverability of convex and visible compacta from their projections, Math. USSR-Sb., 1992, vol. 73, issue 1, pp. 1-10. DOI: 10.1070/SM1992v073n01ABEH002531 Ushakov V.N., Uspenskii A.A. $\alpha$-sets in finite dimensional Euclidean spaces and their properties, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2016, vol. 26, issue 1, pp. 95-120 (in Russian). DOI: 10.20537/vm160109 Golubyatnikov V.P., Rovenski V.Yu. Some extensions of the class of $k$-convex bodies, Sib. Math. J., 2009, vol. 50, issue 5, pp. 820-829. DOI: 10.1007/s11202-009-0092-6 Full text