TY - JOUR
T1 - On The Zeros of Some Complex harmonic polynomials
AU - Mayya, Adithya
AU - Verma, Sarika
AU - Kumar, Raj
AU - Prasad, Kuncham Syam
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag Italia S.r.l., part of Springer Nature 2024.
PY - 2025/2
Y1 - 2025/2
N2 - It is well known that the Fundamental Theorem of Algebra doesn’t hold true in the case of complex harmonic polynomials. The motive of the manuscript is to explore the zeros of a complex harmonic polynomial F(z) of degree (n+m). In particular, we prove that the sum of the orders of the zeros of F(z) is -(n+m). We also show that F(z) has a total (n+m+2) or (n+m+2k+2) number of zeros under different conditions on the coefficients. In addition to the count of zeros, we locate the zeros of F(z) and obtain that all non-trivial zeros of F(z) lie in the given annular region.
AB - It is well known that the Fundamental Theorem of Algebra doesn’t hold true in the case of complex harmonic polynomials. The motive of the manuscript is to explore the zeros of a complex harmonic polynomial F(z) of degree (n+m). In particular, we prove that the sum of the orders of the zeros of F(z) is -(n+m). We also show that F(z) has a total (n+m+2) or (n+m+2k+2) number of zeros under different conditions on the coefficients. In addition to the count of zeros, we locate the zeros of F(z) and obtain that all non-trivial zeros of F(z) lie in the given annular region.
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U2 - 10.1007/s12215-024-01176-3
DO - 10.1007/s12215-024-01176-3
M3 - Article
AN - SCOPUS:85213524930
SN - 0009-725X
VL - 74
JO - Rendiconti del Circolo Matematico di Palermo
JF - Rendiconti del Circolo Matematico di Palermo
IS - 1
M1 - 19
ER -