Marden 's theorem gives a geometric connection between the zeros of the complex polynomial of the third degree and the zeros of its derivative :
Marden's theorem
Suppose that the zeros z 1 , z 2 , z 3 of the polynomial third degree noncollinear . There is a single ellipse inscribed in a triangle with vertices z 1 , z 2 , z 3 and touching its sides in the middle: the Steiner ellipse . The tricks of this ellipse are the zeros of the derivative . |
Marden attributes the theorem to Jörg Siebeck ( German: Jörg Siebeck ) [1] and gives 9 references to articles that include versions of this theorem.
Notes
- ↑ Siebeck, Jörg (1864), Über eine neue analytische Behandlungweise der Brennpunkte, Journal für die reine und angewandte Mathematik T. 64: 175-182, ISSN 0075-4102 (German)
Links
- Badertscher, Erich A simple direct proof of Marden's theorem. Amer. Math. Monthly 121 (2014), no. 6, 547-548.
- Kalman, Dan (April 2008), "An Elementary Proof of Marden's Theorem", The American Mathematical Monthly T. 115: 330–338, ISSN 0002-9890 (eng.)
- Kalman, Dan (April 2008), " The Most Marvelous Theorem in Mathematics ", Journal of Online Mathematics and its Applications (eng.)
- Marden, Morris (1945), " A note on the zeroes of the sections of a partial fraction ", Bulletin of the American Mathematical Society Vol. 51 (12): 935–940, ISSN 0002-9904 , < http: // www .ams.org / bull / 1945-51-12 / S0002-9904-1945-08470-5 / home.html > (eng.)
- Marden, Morris (1966), Geometry of Polynomials , Mathematical Surveys, number 3, Providence, RI: American Mathematical Society (eng.)