*Appendix A.2. Bounds of Real Roots of Polynomials*

The first result in the theory of the location of polynomial zeros is due to Gauss, which is improved by Cauchy in [39], where he proves the following theorem.

**Theorem A1** (**Cauchy**)**.** *Let*

$$f(\mathbf{x}) = a\_n \mathbf{x}^n + a\_{n-1} \mathbf{x}^{n-1} + \dots + a\_1 \mathbf{x} + a\_0 \mathbf{x}$$

*be a polynomial with complex coefficients, where n* ≥ 1 *and an* = 0*. Then, all the zeros of f*(*x*) *lie inside the circle of radius*

$$R = 1 + \max\_{0 \le k \le n-1} \left| \frac{a\_k}{a\_n} \right|$$

*about the origin.*

Another bound given by Lagrange is: Let

$$f(\mathbf{x}) = a\_n \mathbf{x}^n + a\_{n-1} \mathbf{x}^{n-1} + \dots + a\_1 \mathbf{x} + a\_0$$

be a polynomial with complex coefficients, where *n* ≥ 1 and *an* = 0. Then, all the zeros of *f*(*x*) lie inside the circle of radius

$$R\_n = 2 \max\left( \left| \frac{a\_{n-1}}{a\_n} \right|, \left| \frac{a\_{n-2}}{a\_n} \right|^{1/2}, \dots, \left| \frac{a\_0}{a\_n} \right|^{1/n} \right)$$

about the origin.

The next theorem is about bounding positive real roots of polynomials with real coefficients due to Cauchy.

#### **Theorem A2** (**Cauchy**)**.** *Let*

$$f(\mathbf{x}) = a\_n \mathbf{x}^n + a\_{n-1} \mathbf{x}^{n-1} + \dots + a\_1 \mathbf{x} + a\_0 \mathbf{x}$$

*be a polynomial with real coefficients, where n* ≥ 1 *and an* > 0 *and which has s* > 0 *strictly negative coefficients. Then, every positive real root of f*(*x*) *is no larger than r:*

$$R\_- = \max\left( \left| s \frac{a\_{n-1}}{a\_n} \right|^{1/i} : 1 \le i \le n \text{ and } a\_{n-i} < 0 \right).$$

More recent and sharper results are obtained by Joyal, Labelle, and Rahman [40] by proving.

**Theorem A3.** *If M* = max0≤*i*<*n*−<sup>1</sup> |*ai*|*, then all the zeros of the monic polynomial*

$$f(\mathbf{x}) = \mathbf{x}^n + a\_{n-1}\mathbf{x}^{n-1} + \dots + a\_1\mathbf{x} + a\_0\mathbf{x}$$

*are contained in the disc*

$$|\mathbf{x}| \le \frac{1}{2} \left( 1 + |a\_{n-1}| + \sqrt{(1 - |a\_{n-1}|)^2 + 4M} \right).$$

#### **References**


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