A real polynomial 
is said to be stable if all its roots
lie in the left half-plane. The term "stable"
is used to describe such a polynomial because, in the theory of linear servomechanisms,
a system exhibits unforced time-dependent motion of the form 
, where 
is the root of a certain real
polynomial 
.
A system is therefore mechanically stable iff 
is a stable polynomial.
The polynomial 
is stable iff 
, and the irreducible
polynomial 
is stable iff both 
and 
are greater than zero. The Routh-Hurwitz
theorem can be used to determine if a polynomial is stable.
Given two real polynomials 
and 
, if 
and 
are stable, then so is their product 
, and vice versa (Séroul 2000, p. 280). It therefore
follows that the coefficients of stable real polynomials are either all positive
or all negative (although this is not a sufficient
condition, as shown with the counterexample 
). Furthermore, the values of a stable polynomial
are never zero for 
and have the same sign as the coefficients of the polynomial.
It is possible to decide if a polynomial is stable without first knowing its roots using the following theorem due to Strelitz (1977). Let 
be a real polynomial with roots
,
..., 
,
and construct 
as the monic real polynomial of degree 
having roots 
for 
. Then 
is stable iff all coefficients of
and 
are positive (Séroul 2000, p. 281).
For example, given the third-order polynomial 
, the sum-of-roots polynomial 
is given by
 |
(1)
|
Resolving the inequalities given by requiring that each coefficient of 
and 
be greater than zero then gives the conditions for 
to be stable as 
, 
, 
.
Similarly, for the fourth-order polynomial 
, the sum-of-roots-polynomial is
 |
(2)
|
so the condition for 
to be stable can be resolved to 
, 
, 
, 
.
The fifth-order polynomial is
 |
(3)
|
The following Wolfram Language code computes the sum-of-roots polynomial 
and inequalities obtained from the coefficients:
RootSumPolynomial[r_List, x_]:=Module[
{n = Length[r], i, j},
RootReduce@Collect[Expand[
Times@@((x - #)&/@Flatten[
Table[r[[i]] + r[[j]], {i, n},
{j, i+1, n}]])
], x]
]
RootSumPolynomial[p_?PolynomialQ, x_]:=
RootSumPolynomial[RootList[p, x], x]
RootList[p_?PolynomialQ, x_]:=
x /. {ToRules[Roots[p==0, x,
Cubics -> False, Quartics -> False
]]}
RootSumInequalities[p_?PolynomialQ, x_]:=
And @@ (# > 0& /@
Flatten[CoefficientList[#, x]& /@
{RootSumPolynomial[p, x], p}])
while the following reduces the inequalities to a minimal set in the cubic case:
Resolve[Exists[x, Element[(a | b | c | x), Reals],
RootSumInequalities[x^3 + a x^2 + b x + c, x]
], {a, b, c}]
