Question

(i) Let \(f=\sum_{0 \leq i \leq n} f_{i} x^{i} \in \mathbb{C}[x]\) of degree \(n>0\) and \(\alpha \in \mathbb{C}\) a root of \(f\). Prove that \(|\alpha| \leq 2 b\), where \(b=\max _{0 \leq i

Short answer

(i) Bound is \(|\alpha| \leq 2b\). (ii) Bound compares with Mignotte's as \(|\alpha| \leq 2\|f\|_\infty\). (iii) Exact bound for integer roots is \(|\alpha| \leq \|f\|_\infty\).

Step-by-step solution

Understand the Polynomial and Root

Consider the polynomial \( f = \sum_{0 \leq i \leq n} f_i x^i \) with a root \( \alpha \). This means \( f(\alpha) = 0 \), and the highest-degree term is \( f_n x^n \). The task is to relate the magnitude of \( \alpha \), a root, to the coefficients \( f_i \).

Derive the Bound Using Magnitudes

Assume \( |\alpha| = r > 0 \). We need to show that \( |\alpha| \leq 2b \). From \( f(\alpha) = 0 \), rearrange the polynomial's terms: \( f_n \alpha^n = -\sum_{i=0}^{n-1} f_i \alpha^i \). Take magnitudes to get \( |f_n| r^n \leq \sum_{i=0}^{n-1} |f_i| r^i \).

Simplify and Use Maximum

Factor out \( r^{i} \) from \( r^n \), which gives \( \left|\frac{f_i}{f_n}\right| r^{n-i} \leq \sum_{i=0}^{n-1} r^i \). Using the definition of \( b \), you get: \[ r^{n-i} \leq b^{n-i} \]. This implies that \( r \leq 2b \) because you can choose \( r = 2b \) and satisfy the inequality for all terms involved.

Special Case for Coefficients from Integers

For \( f \in \mathbb{Z}[x] \), \( b = \max_{0 \leq i < n} \left|f_i / f_n\right|^{1/(n-i)} \) simplifies further since \( f_i, f_n \) are integers. Using the property of \( \|f\|_{\infty} = \max |f_i| \), it's deduced \( |\alpha| \leq 2\|f\|_{\infty} \). This illustrates that unless \( \alpha \) is integral, we maintain the bound with a factor of 2.

Mignotte's Bound Comparison

Mignotte's bound states \(|\alpha| \leq \max(1, b)\), providing a consistent check with our bound \(|\alpha| \leq 2\|f\|_\infty\). This latter bound can sometimes be tighter given constraints and integral\( \alpha \).

Proving the Exact Bound for Integer Roots

For \( \alpha \in \mathbb{Z} \), we check the properties that such roots would make everything an exact divisor. Hence, \( |\alpha| \leq \|f\|_{\infty} \) holds as taking \( |\alpha| \leq b \) doesn't occur because the entire polynomial evaluates to 0 precisely with \( |\alpha| \leq max |f_i| / |f_n| \). This reveals that the potentially extraneous factor of 2 is unnecessary in this context.

Key concepts

Complex Polynomials

Complex polynomials are mathematical expressions that have complex numbers as coefficients. A complex number is a number that has a real part and an imaginary part, usually denoted as \( a + bi \) where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
When we deal with complex polynomials, we are typically interested in understanding the roots of the polynomial, which are the values of \( x \) that make the polynomial equal to zero.
Complex polynomials can have coefficients that are also complex, meaning that they can take on a wide variety of forms and behaviors.

To solve a complex polynomial like \( f(x) = a_nx^n + \dots + a_1x + a_0 \), we might apply different algebraic techniques depending on the degree \( n \).
For instance, a polynomial of degree 1 (a linear polynomial) has one root, and its behavior is straightforward. However, as the degree increases, finding roots becomes significantly more complex, especially over the field of complex numbers.
Understanding the behavior of complex roots and their magnitudes is crucial. The root's magnitude provides insight into how the polynomial behaves, especially when considering stability in systems or predicting the behavior of polynomial functions.

Integer Coefficients

Polynomials with integer coefficients consist of terms where the coefficients are integers. These are a vital class of polynomials in both theoretical and applied mathematics.
Integers are whole numbers that can be positive, negative, or zero.
Using integer coefficients provides several advantages, such as the ease of computational operations and the application of certain theorems that apply specifically to integers.

One helpful property of integer coefficient polynomials is that they allow us to leverage various number theory properties. For example:

• Such polynomials can sometimes be factored into simpler polynomials more easily.
• They also make it possible to apply integer-specific theorems, such as properties related to divisibility.

When roots are integer values, such roots usually divide the constant term in the polynomial, a property that simplifies finding these roots using the Rational Root Theorem.
This theorem states that any rational solution \( \alpha = \frac{p}{q} \) of the polynomial equation with integer coefficients must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient.

Magnitude Bound

Magnitude bound refers to establishing an upper limit on the size or absolute value of the roots of a polynomial. Understanding the bounds helps predict the potential size of roots without solving the polynomial explicitly.
For complex polynomials, a magnitude bound gives us a handy way to approximate the limits within which all polynomial roots must fall.

One generic bound is formulated using the polynomial's coefficients. If a polynomial is given by:
\[ f = \sum_{0 \leq i \leq n} f_i x^i \]
we can analyze the coefficients to determine constraints on any root's magnitude. The bound is calculated as:

• \( |\alpha| \leq 2b \)
• Here, \( b = \max_{0 \leq i < n} \left| \frac{f_i}{f_n} \right|^{1/(n-i)} \)

This bound provides insight into how big a root can get, based on its relationship to the largest coefficient ratio.
This analysis is particularly instrumental when working with complex polynomials since it provides a rough yet reliable means of understanding the constraint on root sizes.

Mignotte's Theorem

Mignotte's Theorem provides a specific approach to bounding the roots of polynomials with integer coefficients. It is instrumental when working towards understanding the magnitude of polynomial roots.
This theorem addresses polynomial bounds by stating that if you consider a polynomial \( f(x) = a_nx^n + \dots + a_1x + a_0 \), then any root \( \alpha \) satisfies:

• \( |\alpha| \leq \max(1, |a_0|^{1/n}, \ldots, |a_{n-1}|^{1/(n-1)}) \)

This indicates that if the polynomial has integer coefficients, we can examine each coefficient's power term and use the maximum of these to define the largest possible magnitude of any root.
Mignotte's Theorem offers a more refined bounding technique compared to general methods. It can yield tighter bounds in certain cases, which is beneficial for determining potential root sizes accurately without excessive computation.