Local behaviour of polynomials

In this paper we study the local behaviour of a trigonometric polynomial $t(\theta )\,:=\,\sum _{\nu =-n}^{n}\,a_{\nu }\,e^{{i}\nu \theta }$ around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in $[0 \,,\, 2 \pi )$. The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form $\sum _{\nu =0}^{n}\,a_{\nu }\,x^{\nu }$ are also proved.