About a certain relation between two polynomials of the same degree

• Autorzy: Piwowarczyk T.
• Języki publikacji: EN

Abstrakty

EN

This article presents several different methods for solving the problem of how to find a certain relation defined in chapter 2. The first method deals with the identities known in the theory of symmetric polynomials as the elements of a certain vector space. The second method is designed around the matrix transformations between symmetric polynomials. The third method is designed around the property of a linear operator and its characteristic polynomial. The fourth method is designed in the area of complex numbers, and introduces the multiplication group of 'complex roots of one'. Significant improvement in the third and fourth method is made by introducing so called 'block method'. It facilitates all calculations by making them much shorter. The article ends with an example showing symmetry and regularity of all procedures. Finally, the article shows how to solve the problem for any degree n of the polynomial, and for any degree k. At the end of the paper solutions for n < 5 and k < 5 are tabulated.

Słowa kluczowe

EN

roots of polynomials; symmetric polynomials; companion matrix; eigenvalues; transformations

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2011
• Tom: Vol. 21, no. 1
• Strony: 85--103
• Opis fizyczny: Bibliogr. 14 poz., rys., tab.

Twórcy

autor

Piwowarczyk T.

• Faculty of Electrical and Computer Engineering, Cracow Uniwersity of Technology, Kraków 31-155, Warszawska 24

Bibliografia

• [1] J. AMBROSIEWICZ: About certain generalization of associated matrix. ZN WSP, Katowice, 1967, (in Polish).
• [2] S. BELERT and H. WOŹNIACKI: Analisis and synthesis of electrical systems using structural numbers method. WNT, Warsaw, 1968, (in Polish).
• [3] D. Cox, J. LITTLE and D. O’SHEA: Ideals, varieties and algorithms, An introduction to computational algebraic geometry and commutative algebra. Springer New York, 1998.
• [4] A. CAYLEY: The Collected Mathematical Papers. XII Cambridge, 1897.
• [5] J. FLACHSMEYER: Kombinatorik, Eine Einfuerung in die mengentcoretische Denkweise. VEB Deutscher Verlag der Wissenschaften, Berlin, 1969.
• [6] A. KOUROSH: Higher algebra. Mir Publisher, Moscow, 1972.
• [7] B. NOBLE, J.W.DANIEL: Applied Linear Algebra. Prentice-Hall, Englewood Cliffs, New Jersey. 1977
• [8] P. NOWACKI, L. SZKLARSKI and H. GORECKI: Principles of control system theory. PWN, Warszawa, 1958, (in Polish).
• [9] Z. OPIAL: Higher algebra. PWN, Warszawa, 1969, (in Polish).
• [10] T. PIWOWARCZYK: Simplification of rational functions. Cracow University of Technology, Z 17 Cracow, (2008).
• [11] T. PIWOWARCZYK: Multipower notation of symmetric polynomials in engineering calculus. PAN, Krakow, 2000.
• [12] T. PIWOWARCZYK: The theory of finite numerical sequences in the analysis of electrical circuits. CCNS, Cracow, 2002, (in Polish).
• [13] T. PIWOWARCZYK: Symmetric polynomials of several variables in electric circuits. Cracow University of Thechnology, Cracow, 2002, (in Polish).
• [14] A. TUROW1CZ: The geometry of zeros of polynomials. PWN, Warsaw, 1967, (in Polish).