##
**Structure of discriminant set of real polynomial.**
*(Russian.
English summary)*
Zbl 1441.13070

Summary: The problem of description the structure of the discriminant set of a real polynomial often occurs in solving various applied problems, for example, for describing a set of stability of stationary points of multiparameter systems, for computing the normal form of a Hamiltonian system in vicinity of equilibrium in the case of multiple frequencies. This paper considers the structure of the discriminant set of a polynomial with real coefficients. There are two approaches to its study. The first approach is based on the study of zeros of ideals formed by the set of subdiscriminants of the original polynomial. Different ways of computing subdiscriminants are given. It is proposed to investigate the singular points of the discriminant set in the second approach. By the methods of computer algebra it is shown that for small values of the degree of the original polynomial, both approaches are equivalent, but the first one is preferred because of smaller ideals.

A constructive algorithm is proposed for obtaining polynomial parameterization of the discriminant set in the space of coefficients of the polynomial. From the applied point of view the most interesting is the description of the components of codimension 1 of the discriminant set. It is this component that divides the space of the coefficients into the domains with the same structure of the roots of the polynomial. The set of components of different dimensions of the discriminant set has a hierarchical structure. Each component of higher dimensions can be considered as some kind of tangent developable surface which is formed by linear varieties of respective dimension. The role of directrix of this component performs a variety of dimension one less than that on which the original polynomial has only multiple zero and the remaining zeroes are simple.

Starting with a one-dimensional algebraic variety of dimension 1 on which the original polynomial has the unique zero of maximal multiplicity, in the next step of the algorithm we obtain the description of the variety on which the polynomial has a pair of zeros: one simple and another multiple. Repeating sequentially the steps of the algorithm, the resulting parametric representation of components of codimension 1 of the discriminant set can be obtained.

Examples of the discriminant set of a cubic and quartic polynomials are considered.

A constructive algorithm is proposed for obtaining polynomial parameterization of the discriminant set in the space of coefficients of the polynomial. From the applied point of view the most interesting is the description of the components of codimension 1 of the discriminant set. It is this component that divides the space of the coefficients into the domains with the same structure of the roots of the polynomial. The set of components of different dimensions of the discriminant set has a hierarchical structure. Each component of higher dimensions can be considered as some kind of tangent developable surface which is formed by linear varieties of respective dimension. The role of directrix of this component performs a variety of dimension one less than that on which the original polynomial has only multiple zero and the remaining zeroes are simple.

Starting with a one-dimensional algebraic variety of dimension 1 on which the original polynomial has the unique zero of maximal multiplicity, in the next step of the algorithm we obtain the description of the variety on which the polynomial has a pair of zeros: one simple and another multiple. Repeating sequentially the steps of the algorithm, the resulting parametric representation of components of codimension 1 of the discriminant set can be obtained.

Examples of the discriminant set of a cubic and quartic polynomials are considered.

### MSC:

13P15 | Solving polynomial systems; resultants |

68W30 | Symbolic computation and algebraic computation |

PDF
BibTeX
XML
Cite

\textit{A. B. Batkhin}, Chebyshevskiĭ Sb. 16, No. 2(54), 23--34 (2015; Zbl 1441.13070)

### References:

[1] | Batkhin A. B., Bruno A. D., Varin V. P., “Stability sets of multiparameter Hamiltonian systems”, J. Appl. Math. Mech., 76:1 (2012), 56-92 · Zbl 1272.70076 |

[2] | Batkhin A. B., “Stability of the certain multiparameter Hamiltonian system”, Preprinty IPM, 2011, 069 |

[3] | Gryazina E. N., Polyak B. T., Tremba A. A., “\(D\)-decomposition technique state-of-the-art”, Automation and Remote Control, 69:12 (2008), 1991-2026 · Zbl 1160.93003 |

[4] | Markeev A. P., Libration Points in Celestial Mechanics and Cosmodynamics, Nauka, M., 1978 · Zbl 1454.70002 |

[5] | Neiman N. N., “Some problems on the distributions of the zeroes of polynomials”, Uspekhi Mat. Nauk, 4:6(34) (1949), 154-188 (in Russian) |

[6] | Basu S., Pollack R., Roy M.-F., Algorithms in Real Algebraic Geometry, Springer-Verlag, Berlin-Heidelberg-New York, 2006 · Zbl 1102.14041 |

[7] | Kalinina E. A., Uteshev A. Yu., Elimination theory, Izd-vo NII Khimii SPbGU, Saint-Petersburg, 2002 |

[8] | Sylvester J. J., “On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm”s function”, Trans. Roy. Soc. London, 1853 |

[9] | Bézout É., Théorie générale des Équations Algébrique, P.-D. Pierre, Paris, 1779 · Zbl 1129.12001 |

[10] | Habicht W., “Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens”, Comm. Math. Helvetici, 21 (1948), 99-116 · Zbl 0029.24402 |

[11] | Uteshev A. Yu., Cherkasov T. M., “The search for the maximum of a polynomial”, J. Symbolic Computation, 25:5 (1998), 587-618 · Zbl 0915.65062 |

[12] | Jury E., Inners and stability of dynamic systems, John Wiley and Sons, 1974 · Zbl 0307.93025 |

[13] | Oprea J., Differential Geometry and its Applications, The Mathematical Assosiation of America, 2007 |

[14] | Finikov S. P., Theory of Surfaces, GTTI, M., 1934 |

[15] | Cox D., Little J., O’Shea D., Ideals, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997 · Zbl 0861.13012 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.