Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 5 / 1 / Solution

The Markov interlacing lemma says that differentiation preserves interlacing roots of polynomials. More explicitly, if two degree-$d$ real polynomials have ordered real roots satisfying $a_1\le b_1\le a_2\le \cdots \le a_d\le b_d$, their $k$th derivatives have ordered roots $a_j^{(k)},b_j^{(k)}$ satisfyingThe analogous alternating order is preserved for polynomials of consecutive degrees. Common roots are allowed through weak inequalities; strictly interlacing simple roots give strict inequalities. When a derivative is constant, its root list is empty.