A BOUND FOR THE SQUARE OF THE ZEROS

José Luis Diaz-Barrero

Abstract. In this note we use matrix methods for obtaining an explicit bound for the moduli of the square of the zeros of a polynomial.

1. Introduction. It is well–known that matrix methods can be used to obtain certain root–location theorems for polynomial equations. That the zeros of the monic polynomial

A_{n}(z) = a_{0} + a_{1}z + ⋯ + a_{n−1}z^{n−1} + z^{n},

are the eigenvalues of its companion matrix leads to easier methods in obtaining bounds for the zeros of polynomials [1,2]. Among others, Aziz and Mohammad [3], by making use of Gershgorin’s Theorem [4], have derived some bounds as a direct consequence of the above fact. In this note we apply Gershgorin’s result to obtain, as far as we know, a new explicit bound for the squares of the zeros of A_{n}(z).

\underline{Theorem 1. (Gershgorin)}. Let A = (a_{ij}) be an n × n complex matrix, and let R_{i} be the sum of the moduli of the off–diagonal elements in the ith row. Then each eigenvalue of A lies in the union of the circles

|z − a_{ii}| ≤ R_{i}, i = 1, 2, …, n.

The analogous result holds if columns of A are considered.

2. The Main Result. In what follows an explicit expression of the coefficients of the polynomials whose zeros are the squares of those of A_{n}(z) in terms of their coefficients is given, and one theorem on location of the zeros is proved. It can be stated as the following theorem.

\underline{Theorem 2}. Let z_{1}, z_{2}, …, z_{n} be the zeros of the monic complex polynomial

A_{n}(z) = ∑_{k=0}^{n} a_{k}z^{k}.

Then, z_{i}^{2}, i = 1, 2, …, n lies in the disk C = {z ∈ ℂ : |z| ≤ r}, where

r = 2 max_{0≤k≤n−1} \{\frac{|a_{k}|^{2} + ∑_{j=1}^{n−k+1} |a_{k−j}a_{k+j}|}{|a_{k+1}|^{2}}\}

with a_{l} = 0 if l < 0 or l > n.

\underline{Proof}. The preceding statement is a consequence of Gershgorin’s result. In order to prove it, we will use the larger circles

|z| ≤ |a_{ii}| + R_{i}, i = 1, 2, …, n,

instead of the ones given in Theorem 1. Let

B_{n}(z) = ∑_{k=0}^{n} b_{k}z^{k}

be the monic complex polynomial whose zeros are z_{1}^{2}, z_{2}^{2}, …, z_{n}^{2} and let

F (B_{n}) = ( \begin{matrix}0 & 0 & ⋯ & 0 & −b_{0} \\ 1 & 0 & ⋯ & 0 & −b_{1} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & −b_{n−1} \end{matrix} )

be its companion matrix. Setting

D = diag(|a_{1}|^{2}, |a_{2}|^{2}, …, |a_{n−1}|^{2}, |a_{n}|^{2}),

we form the matrix

D^{−1}F D = ( \begin{matrix}0 & 0 & ⋯ & 0 & −b_{0}/|a_{1}|^{2} \\ \frac{|a_{1}|^{2}}{|a_{2}|^{2}} & 0 & ⋯ & 0 & −b_{1}/|a_{2}|^{2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & \frac{|a_{n−1}|^{2}}{|a_{n}|^{2}} & −b_{n−1}/|a_{n}|^{2} \end{matrix} ) .

Since the eigenvalues of D^{−1}F D are the same as those of F, i.e., the zeros of B_{n}(z); by direct application of Theorem 1, we have

|z_{i}^{2}| ≤ max_{1≤k≤n−1} \{\frac{|b_{0}|}{|a_{1}|^{2}} , \frac{|a_{k}|^{2}}{|a_{k+1}|^{2}} + \frac{|b_{k}|}{|a_{k+1}|^{2}} \}, i = 1, 2, …, n.  (2.1)

On the other hand, the coefficients of B_{n}(z) are related to coefficients of A_{n}(z) by the expressions [5]:

b_{k} = (−1)^{n−k}(a_{k}^{2} + 2 ∑_{j=1}^{n−k+1}(−1)^{j}a_{k−j}a_{k+j}), k = 0, 1, …, n

with a_{l} = 0 if l < 0 or l > n. Substituting (2.2) into (2.1), we have

|z_{i}^{2}| ≤ max_{1≤k≤n−1} \{\frac{|b_{0}|}{|a_{1}|^{2}} , \frac{|a_{k}|^{2} + |b_{k}|}{|a_{k+1}|^{2}}\}

= max_{1≤k≤n−1} \{\frac{|a_{0}|^{2}}{|a_{1}|^{2}} , \frac{|a_{k}|^{2} + |(−1)^{n−k}(a_{k}^{2} + 2 ∑_{j=1}^{n−k+1}(−1)^{j}a_{k−j}a_{k+j})|}{|a_{k+1}|^{2}}\}

≤ max_{1≤k≤n−1} \{\frac{|a_{0}|^{2}}{|a_{1}|^{2}} , \frac{2|a_{k}^{2}| + 2 ∑_{j=1}^{n−k+1} |a_{k−j}a_{k+j}|}{|a_{k+1}|^{2}}\}

≤ max_{0≤k≤n−1} \{\frac{|a_{k}^{2}| + ∑_{j=1}^{n−k+1} |a_{k−j}a_{k+j}|}{|a_{k+1}|^{2}}\}, i = 1, 2, …, n.

Note that the equality holds only when a_{0} = 0.

Finally, we have to prove (2.2). We argue by mathematical induction. The first cases when n = 1, 2, 3 can be checked by inspection. Next, we assume that the following expression holds

B_{n}(z) = ∑_{k=0}^{n}(−1)^{n−k}[a_{k}^{2} + 2 ∑_{j=1}^{n−k+1}(−1)^{j}a_{k−j}a_{k+j}]z^{k}.

We have to prove that

B_{n+1}(z) = z^{n+1} + ∑_{k=0}^{n}(−1)^{n−k+1}[a_{k}^{2} + 2 ∑_{j=1}^{n−k+2}(−1)^{j}a_{k−j}a_{k+j}]z^{k}.

In fact,

B_{n+1}(z) = B_{n}(z)(z − z_{n+1}^{2})

= { ∑_{k=0}^{n}(−1)^{n−k}[a_{k}^{2} + 2 ∑_{j=1}^{n−k+1}(−1)^{j}a_{k−j}a_{k+j}]z^{k}}(z − z_{n+1}^{2})

= ∑_{k=0}^{n}(−1)^{n−k}[a_{k}^{2} + 2 ∑_{j=1}^{n−k+1}(−1)^{j}a_{k−j}a_{k+j}]z^{k+1}

− z_{n+1}^{2}{ ∑_{k=0}^{n}(−1)^{n−k}[a_{k}^{2} + 2 ∑_{j=1}^{n−k+1}(−1)^{j}a_{k−j}a_{k+j}]z^{k}}

= (−1)^{n+1}a_{0}^{2}z_{n+1}^{2} + (−1)^{n}[a_{0}^{2} + z_{n+1}^{2}(a_{1}^{2} + 2(−1)^{1}a_{0}a_{2})]z + ⋯

+ (−1)^{1}[a_{n−1}^{2} + 2(−1)^{1}a_{n−2} + z_{n+1}^{2}]z^{n} + z^{n+1}.  (2.3)

Finally, taking into account that

\begin{aligned}A_{n+1}(z) & = A_{n}(z)(z − z_{n+1}) = z^{n+1} + ∑_{k=1}^{n}(a_{k−1} − a_{k}z_{n+1})z^{k} − a_{0}z_{n+1} \\ & = ∑_{k=0}^{n+1} a_{n+1,k}z^{k}, a_{n+1,n+1} = 1, \end{aligned}

and (2.3), we have

B_{n+1}(z) = (−1)^{n+1}a_{n+1,0}^{2} + (−1)^{n}[a_{n+1,1}^{2} + 2(−1)^{1}a_{n+1,0}a_{n+1,2}]z + ⋯

+ (−1)^{1}[a_{n+1,n}^{2} + 2(−1)^{1}a_{n+1,n−1}]z^{n} + z^{n+1}

= z^{n+1} + ∑_{k=0}^{n}(−1)^{n+1−k}[a_{n+1,k}^{2} + 2 ∑_{j=1}^{n−k+2}(−1)^{j}a_{n+1,k−j}a_{n+1,k+j}]z^{k}

and (2.2) is proved. Note that the first subscript in the preceding expressions has been used for pointing out the polynomial degree. This completes the proof of Theorem 2.

For example, if we consider the polynomial

A(z) = z^{3} − 1.1z^{2} − 1.2z + 1.3,

the squares of all its zeros lie in the disk C = {z ∈ ℂ : |z| < r} where r ≃ 4.82. This bound is sharper than the explicit bound of Cauchy D = {z ∈ ℂ : |z| < 5.3}.

\underline{References}

1. M. Marden, Geometry of Polynomials, American Mathematical Society, Providence, Rhode Island, 1989.

2. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1990.

3. A. Aziz and Q. G. Mohammad, “Zero Free–Regions for Polynomials and Some Generalizations of Enerströn–Kakeya Theorem,” Canad. Math. Bull., 27 (1984), 265–272.

4. S. Gershgorin, “Uber die Abgrenzung der Eigenwerte einer Matrix,” Izv. Akad. Nauk SSSR, 7 (1931), 749–754.

5. J. L. Diaz, “Answer to Problem 2388,” CRUX Mathematicorum with Mathematical Mayhem, 25 (1999), 445.

José Luis Diaz-Barrero Applied Mathematics III Universitat Politècnica de Catalunya Jordi Girona 1–3, C–2 08034 Barcelona Spain email: jose.luis.diaz@upc.es