application/pdf 364410 bytes Comptes rendus de l’Académie bulgare des Sciences volume: 77 number: 3 startpage: 325 endpage: 329 Abstract We are proving Coincidence theorem due to Walsh for the case when the total degree of a polynomial is less than the number of arguments. Also, the following result has been proven: if p(z) is a complex polynomial of degree n, then closed disk D that contains at least n−1 of its zeros (counting multiplicity) contains at least [ n − 2k + 1 2 ] zeros of its k-th derivative, provided that the arithmetical mean of these zeros is also centre of D. We also prove a variation of the classical composition theorem due to Szeg¨o. 2024 Bulgarian Academy of Sciences eng info:eu-repo/semantics/article https://phaidrabg.bg.ac.rs/o:37488 doi:10.7546/CRABS.2024.03.01 ISSN: 1310–1331 Bakić, Radoš All rights reserved Key words: Coincidence theorem, zeros of polynomial, critical points of a polynomial, apolar polynomials 2020 Mathematics Subject Classification: Primary 26C10, Secon- dary 30C15 On the Coincidence Theorem