Definition 4.25 (Common interlacement). We say that real rooted polynomials $f$, $g$ of degree $n$ have a common interlacement if there is a polynomial $h$ of degree $n+1$ such that $f$ and $g$ both interlace $h$. In other words, if the roots of $f$ and $g$ are ${\alpha }_1\le \cdots {\alpha }_n$ and ${\beta }_1\le \cdots \le {\beta }_n$ respectively, then they have a common interlacement if and only if there are some ${\gamma }_0\le \cdots \le {\gamma }_n$ such that