Geodesic
On the Zeros of Apolar Polynomials
Kragujevac Journal of Mathematics, Tome 49 (2025) no. 4, p. 653

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The classical notion of apolarity is defined for two complex polynomials of same degree. The main property of two apolar polynomials, $f(z)$ and $g(z)$ was given by Grace's theorem which states that ``every circular domain containing all the zeros of $f(z)$ contains at least one zero of $g(z)$ and vice-versa''. A. Aziz [1] dropped the condition that $f(z)$ and $g(z)$ are of the same degree in case the circular domain is a disk. In this paper, we extend the result of A. Aziz to every kind of circular domain and hence an extension of Grace's theorem for two arbitrary polynomials is proved. This also allows us to generalise the results of Walsh, Szego, Takagi, Aziz and several other results about apolar polynomials.
Classification : 30C10, 30C15
Keywords: polynomial, apolar, zeros, circular domain 
Ishfaq Nazir; Mohammad Ibrahim Mir; Irfan Ahmad Wani. On the Zeros of Apolar Polynomials. Kragujevac Journal of Mathematics, Tome 49 (2025) no. 4, p. 653 . http://geodesic.mathdoc.fr/item/KJM_2025_49_4_a10/
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     title = {On the {Zeros} of {Apolar} {Polynomials}},
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     year = {2025},
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     url = {http://geodesic.mathdoc.fr/item/KJM_2025_49_4_a10/}
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