roots of polynomials outside the unit disc

Is there any "sufficient and necessary" condition for a comlpex polynomial P(z)=a_0+...a_n z^n, in terms of its coefficinets, to have all its roots outside the unit disc(or equivalently inside the unit disc)?

If not, is there any such conditions where the coefficinets are real or rational or integers?

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The "theorem of Burnside" is trivial, no? The "roots on the unit circle" result is usually attributed to Kronecker. –  Gerry Myerson Jun 6 '11 at 23:20

Look up the Schur-Cohn algorithm. It's discussed at length in Volume I of Henrici's "Applied and Computational Complex Analysis, and its exactly what you need.

It runs as follows. If $p$ is a complex polynomial of degree $n$ with leading coefficient $a_0$ and constant term $a_n$, define $T(p)$ to be the polynomial $$\bar{a}_0 p(z) -a_np^*(z).$$

(Here $p^*(z)$ is the reciprocal polynomial $z^np(1/z)$.) Then all zeros of $p$ lie outside the closed unit disc if and only if the numbers $T^k(p)(0)$ for $k=1,\ldots,n$ are all positive.

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Thank you. seems great. –  mosen Jun 8 '11 at 9:53