Primordial polynomials
Source: India IMOTC 2024 Day 4 Problem 1
May 31, 2024
Polynomialsalgebrapolynomial
Problem Statement
Let be a real number. We call a monic polynomial with complex coefficients -good if all of its roots have absolute value at most . We call a monic polynomial with complex coefficients primordial if all of its coefficients have absolute value at most .
a) Prove that any -good polynomial has a primordial multiple.
b) If , prove that there exists an -good polynomial that does not have a primordial multiple.Proposed by Pranjal Srivastava