Stable Polynomial

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}]

REFERENCES:

Séroul, R. "Stable Polynomials." §10.13 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 280-286, 2000.

Strelitz, S. "On the Routh-Hurwitz Problem." Amer. Math. Monthly 84, 542-544, 1977.

Tóth, J.; Szili, L.; and Zachár, A. "Stability of Polynomials." Mathematica Educ. Res. 7, 5-12, 1998.