o:37488 On the Coincidence Theorem en 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. Key words: Coincidence theorem, zeros of polynomial, critical points of a polynomial, apolar polynomials 2020 Mathematics Subject Classification: Primary 26C10, Secon- dary 30C15 1552099 10.7546/CRABS.2024.03.01 1552101 1310–1331 2026-01-19T11:57:34.916Z 44 yes 46 Radoš Bakić University of Belgrade, Faculty of Education, Belgrade 0000-0002-5280-011X application/pdf 364410 https://phaidrabg.bg.ac.rs/o:37488 no yes 1 70 92000001 11A42 Comptes rendus de l’Académie bulgare des Sciences 77 3 325 329 Bulgarian Academy of Sciences 2024 https://www.proceedings.bas.bg/index.php/cr/article/view/493