# Dynamic Geometry of Some Polynomials

## Abstract

In 1836, Gauss showed that all the roots of P0, distinct from the multiple roots of the polynomial P itself, serve as the points of equilibrium for the field of forces created by identical particles placed at the roots of P (provided that r particles are located at the root of multiplicity r). The following equality to zeros provides a quick proof of Gauss-Lucas theorem (see for example [1] or [2]). Thus was appeared the branch of mathematics, which after the book of Morris Marden [3], was called Geometry of Polynomials. The polynomial conjectures of Sendov and Smale are two challenging problems of this branch [4, 5, 6].

One of the beautiful theorems of mathematics is Marden's theorem [3, 7]. It gives a geometric relationship between the zeros of a third-degree polynomial with complex coefficients and the zeros of its derivative. A more general version of this theorem, due to Linfield [8].

This article focuses on the dynamic behavior of critical points in the case of moving one of the roots of cubic polynomial on a given trajectory. The equations of the curves, where the critical points moves, are obtained. Discovered new geometric properties of positions of the zeros and critical points of a complex polynomial of degree three. The case of multiple roots of the given polynomial is considered as well.

## References

Prasolov, V.V, Polynomials, Springer, 2000.

Rahman, Q.I. and Schmeisser, G., Analytic theory of polynomials, Oxford Univ. Press, 2005.

Marden M., Geometry of Polinomials, AMS, 1966.

Schmeisser, G., The conjectures of Sendov and Smale, Approximation Theory (a volume dedicated to Blagovest Sendov), Sofia Darba, 2002, pp. 353-369.

Smale S., The fundamental theorem of algebra and complexity theory, Bulletin of AMS 4 (1981), pp. 1-36.

Sendov Bl., Generalization of a conjecture in the Geometry of Polynomials, Serdica Math. J. 28 (2002), pp. 283-304.

Kalman, D., An Elementary Proof of Marden's Theorem, The American Mathematical Manthly, vol. 115, no. 4, 2008, pp. 330-338.

Linfield, B.Z., On the relation of the roots and poles of a rational function to the roots of its derivative, Bulletin of the AMS, 27 (1920), pp. 17-21.

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*Mathematical Problems of Computer Science*,

*38*, 37–37. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/479

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