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### Theorems on Roots of Polynomials in Geometry, Topology, Complex Analysis and Functional Analysis

Roots of Polynomials
Prof. V. Kannan
Dept. of Mathematics
SRM University-AP
23 February 2022

Abstract: &quot;We discuss four classical theorems on roots of polynomials. They relate to four
different branches of Mathematics, namely, Geometry, Topology, Complex Analysis and
Functional Analysis. The theorems are stated below:
Theorem 1 (Lucas). Every root of the derivative of a polynomial lies within the convex
polygon formed by the roots of that polynomial.
Theorem 2: The roots of a polynomial vary continuously with its coefficients.
Theorem 3 (Cauchy): If all the coefficients of a polynomial lie in the unit disc D, then all
the roots lie in the disc 2D.
Theorem 4 (Bernstein): The norm of the linear operator D (namely the differentiation) on
(The space of polynomials of degree at most n on D) is exactly n.&quot;