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_α(x) to each such polynomial p, and define cn(p), il(p) to be the sharp threshold values of α that guarantee that, for all larger values of the parameter, p_α(x) 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 il(p) ≥ cn(p), and this inequality is often used for showing circle rootedness. Both il(p) and cn(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 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 il(p) = cn(p), whereas in general the quotient il(p)/cn(p) is shown to be unbounded as the degree grows. Several examples and open questions are presented. This is joint work with Arnaldo Mandel.

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