In mathematics, Cohn's theorem[1] states that a nth-degree self-inversive polynomial
has as many roots in the open unit disk
[1][2][3] Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the complex plane.
[4][5] An nth-degree polynomial, is called self-inversive if there exists a fixed complex number (
Self-inversive polynomials have many interesting properties.
[6] For instance, its roots are all symmetric with respect to the unit circle and a polynomial whose roots are all on the unit circle is necessarily self-inversive.
The coefficients of self-inversive polynomials satisfy the relations.
is a (n − 1)th-degree polynomial given by Therefore, Cohn's theorem states that both