Problem

Source: India IMOTC 2024 Day 4 Problem 1

Tags: Polynomials, algebra, polynomial



Let $r>0$ be a real number. We call a monic polynomial with complex coefficients $r$-good if all of its roots have absolute value at most $r$. We call a monic polynomial with complex coefficients primordial if all of its coefficients have absolute value at most $1$. a) Prove that any $1$-good polynomial has a primordial multiple. b) If $r>1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple. Proposed by Pranjal Srivastava