Marden's theorem

Triangle and inscribed ellipse to the zeros (black) , and from and zeros (red) , from ${\displaystyle 1+i}$ ${\displaystyle 5+2i}$ ${\displaystyle 3+6i}$ ${\displaystyle p(z)=z^{3}-(9+9i)\cdot z^{2}+(3+52i)\cdot z+(33-39i)}$ ${\displaystyle 3.32+4.05i}$ ${\displaystyle 2.68+1.95i}$ ${\displaystyle p'(z)=3\cdot z^{2}+(-18-18i)\cdot z+(3+52i)}$

The set of Marden (after Morris Marden) is a mathematical theorem in the area of function theory . It describes a geometric relationship between the zeros of a third degree polynomial and the zeros of its derivative . According to Gauss-Lucas theorem , the zeros of the derivative must lie within the triangle formed by the zeros of the polynomial in the complex number plane . Marden's theorem also provides an exact localization and is as follows:

Let be a complex third degree polynomial. If it has three different non- collinear zeros, then the triangle formed by them in the complex number plane has an inscribed ellipse that touches the triangle in the middle of the sides and whose two focal points are the zeros of . ${\displaystyle p(z)}$ ${\displaystyle p(z)}$ ${\displaystyle p'(z)}$

The ellipse appearing in the sentence is the Steiner inellipse of the triangle formed by the roots of the polynomial. Although the phrase is now named after Morris Marden, it was not himself that he discovered it. Marden described the result in an article in 1945 and later in his book Geometry of Polynomials (1966) without using a special name for it. However, he cites a number of earlier publications, beginning with a publication by Jörg Siebeck in Crelles Journal (1864). The presentation of the sentence in an article in the American Mathematical Monthly by mathematician Dan Kalman won the 2008 Lester Randolph Ford Award .

swell

• Jörg Siebeck: About a new analytical way of treating focal points. In: Journal for pure and applied mathematics (Crelle). 1864, Issue 64, pp. 175-182, ISSN  0075-4102 .
• Dan Kalman: The Most Marvelous Theorem in Mathematics. In: Journal of Online Mathematics and its Applications (now Loci - Online Journal of the MAA ). April 2008.
• Dan Kalman: An Elementary Proof of Marden's Theorem. In: The American Mathematical Monthly. 115, April 2008, pp. 330-338, ISSN  0002-9890 .
• Morris Marden: A note on the zeroes of the sections of a partial fraction. In: Bulletin of the American Mathematical Society. 51 (12), pp. 935-940, 1945, doi : 10.1090 / S0002-9904-1945-08470-5 .

Individual evidence

1. ^ Dan Kalman: The Most Marvelous Theorem in Mathematics. In: Journal of Online Mathematics and its Applications (now Loci - Online Journal of the MAA ). April 2008.