Interlacing roots of polynomials ($a_1\le b_1\le a_2\le \cdots \le a_d\le b_d$)

Two real polynomials with only real roots of a polynomial interlace when their ordered roots alternate, counting multiplicity of a root. For equal polynomial degrees, one ordering is the displayed chain; the opposite ordering is also permitted. Consecutive degrees give the analogous chain with one extra root at each end for the higher-degree polynomial. Strict interlacing uses strict inequalities. Weak interlacing permits common roots and follows as a limit of strict interlacing.

For a monic polynomial $p(x)={\prod }_i(x-a_i)$ with distinct ordered real roots of a polynomial, each critical point $\xi$ lies between two consecutive roots by Rolle's theorem. Its logarithmic derivative gives ${\sum }_i(\xi -a_i{)}^{-1}=0$. Implicit differentiation gives the displayed formula, so moving any root to the right moves every ordered critical point to the right, while the roots remain distinct. Iterating this statement for derivatives preserves componentwise ordering of their roots. Weak inequalities extend by continuity to repeated roots.

Differentiation preserves the ordering of interlacing roots of polynomials of equal or consecutive polynomial degrees. Iteration gives the result for every derivative for which both polynomials are nonconstant; common roots are allowed by a limiting argument. This is a result about the locations of roots of a polynomial, distinct from the Markov inequality for polynomial derivatives. Theorem A of Dimitar K. Dimitrov's account states the first-derivative result.