Dragging the roots of a polynomial to the unit circle

Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials $p_\alpha$ to each such polynomial $p$, and define $\mathscr{cn}(p)$, $\mathscr{il}(p)$ to be the sharp threshold values of $\alpha$ that guarantee that, for all larger values of the parameter, $p_\alpha$ has, respectively, all roots in the unit circle and all roots interlacing the roots of unity of the same degree. Interlacing implies circle rootedness, hence $\mathscr{il}(p)\geq\mathscr{cn}(p)$, and this inequality is often used for showing circle rootedness. Both $\mathscr{cn}(p)$ and $\mathscr{il}(p)$ turn out to be semi-algebraic functions of the coefficients of $p$, and some useful bounds are also presented, entailing several known results about roots in the circle. The study of $\mathscr{il}(p)$ leads to a rich classification of real self-inversive polynomials of each degree, organizing them into a complete polyhedral fan. We have a close look at the class of polynomials for which $\mathscr{il}(p)=\mathscr{cn}(p)$, whereas in general the quotient $\frac{\mathscr{il}(p)}{\mathscr{cn}(p)}$ is shown to be unbounded as the degree grows. Several examples and open questions are presented.