# Conditions that Roots of a Polynomial be Less than Unity

Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots be less than unity?

Is there a fundamental result (however primitive) that will connect all of these results? Furthermore the Jury Test or Schur's Criterion-- why do we have all of these 'tests'? Is one the most general, and understanding it will make the rest more clear?

I can't answer your question well but I can point you toward a good reference.

Chapter 7 of Marden's Geometry of Polynomials (1949) is dedicated to bounding the zeros of a polynomial by functions of the coefficients of the polynomial. A central tool in the proofs is indeed Rouché's theorem. In chapter 9 he also considers the problem of finding the number of zeros which lie in a half-plane.

As an aside, I get a lot of use out of the Eneström-Kakeya theorem, a good reference for which is this pdf. See also this question on M.SE. One statement of the theorem goes like

If $0 \leq a_0 \leq a_1 \leq \cdots \leq a_n$ then all zeros of the polynomial $$a_0 + a_1 z + \cdots + a_n z^n$$ lie in the closed unit disk.

In the Samuelson paper you linked he considers the polynomial

$$p(x) = -.027 - .013x + .220x^2 - .398 x^3 + x^4.$$

By the K-E theorem the related polynomial

\begin{align} q(x) &= p(-x) + .027 \\ &= .013x + .220 x^2 + .398x^3 + x^4 \end{align}

has all its roots in $|x| \leq 1$, and thus so does $p(x) + .027$. Now on the circle $|x| = 1$ we have

$$|p(x) + .027| \geq \left|x^4\right| - \left(|-.398|\left|x^3\right| + |.220|\left|x^2\right| + |-.013||x|\right) = .369 > |-.027|$$

by the triangle inequality, so by Rouché's theorem we conclude that $p(x)$ has all its roots in $|x| < 1$.

Routh–Hurwitz stability criterion is a necessary and sufficient criterion for the Hurwitz stability of a polynomial, meaning that the real part of all the roots of the polynomial are negative.

Note that by using a bilinear transform (also called Tustin transform), one can also use the Routh-Hurwitz stability criterion to test the Schur stability of the polynomial, that is, if all the roots are inside the unit circle.

Although using Routh-Hurwitz stability criterion with the bilinear transform is a necessary and suficient test for Schur stability, there are more quick tests (which also are necessary and sufficient) to do so, such as the mentioned Jury and Schur–Cohn tests.

AFAIK, the simplest necessary and sufficient test for Schur stability is the Bistritz stability criterion. It is also interesting to note that for the Hurwitz stability of a polynomial there is also a faster test than Routh-Hurwitz, namely the Lienard-Chipart criterion.