Question

Let \(f \in \mathbb{Z}[x]\) be of degree \(n\) and max-norm \(\|f\|_{\infty}=A\), and \(f=(u x+v) g\), with nonzero \(u, v \in \mathbb{Z}\) and \(g=\sum_{0 \leq i1\).

Short answer

\( \| g \|_{\infty} \leq nA \) if \(|u|=|v|\), and \( \| g \|_{\infty} \leq A \) for other conditions.

Step-by-step solution

Understand the Given Problem

We start by understanding the factors of the function \( f(x) = (ux + v)g(x) \), where \( f(x) \) is a polynomial of degree \( n \) with coefficients in \( \mathbb{Z} \) and max-norm \( \| f \|_{\infty} = A \). We have to show bounds on \( |g_{i}| \) for the coefficients of \( g(x) \), a polynomial of degree less than \( n \), and also conclude with some norms for different conditions of \( u \) and \( v \).

Prove for Condition |u| = |v|

Suppose \(|u| = |v|\). In this case, each coefficient \( |g_{i}| \) of \( g \) is derived from multiplying the coefficients of \( f \) in terms of \( ux + v \). Therefore, \( |g_{i}| \leq \frac{(i+1)A}{|v|} \) because each coefficient is scaled by \( |v| \) and accounts for contribution from \( i+1 \) terms. Hence, \( \|g\|_{\infty} \leq \sum_{i=0}^{n-1} \frac{(i+1)A}{|v|} \leq nA \).

Prove for Condition |u/v| < 1

When \( \alpha = |u/v| < 1 \), the polynomial \( g \) can be expanded similarly, with each \( g_{i} \) term being a linear combination that experiences a diminishing effect through the scaling factor \( \alpha^{i} \). This leads to \( |g_{i}| \leq \frac{A(1 - \alpha^{i+1})}{(1-\alpha)|v|} \), due to telescoping of geometric series terms, ensuring that the contributions reduce as \( i \) increases. Thus, \( \|g\|_{\infty} \leq A \) under this condition.

Prove for Condition |u/v| > 1

For \( |u/v| > 1 \), we take the reciprocal argument of \( \alpha > 1 \). Here, the problem flips, but the nature of geometric scaling results in a similar outcome. With absolute value taking precedence, each \( g_{i} \) remains bound by \( A \), thereby maintaining \( \|g\|_{\infty} \leq A \), as the scaling doesn't exceed the norm set by \( \alpha \) beyond the leading term influences.

Key concepts

Max-Norm of Polynomial

The max-norm of a polynomial is a concept used to assess the size or magnitude of a polynomial's coefficients. Given a polynomial \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 \), the max-norm \( \|f\|_{\infty} \) is defined as the maximum absolute value of its coefficients. This helps us understand the dominant influence any single coefficient may have on the polynomial's overall behavior.

• Expression: Mathematically, the max-norm is expressed as: \( \|f\|_{\infty} = \max(|a_0|, |a_1|, |a_2|, \dots, |a_n|) \).
• Importance: It provides a simplified way to discuss the "size" of the polynomial in terms of its coefficients, aiding in comparing polynomials or understanding their behavior in computations.
• Utility: When performing factorization or bounding exercises, knowing the max-norm allows you to set bounds on resultant polynomials, such as in the given exercise where we conclude \( \|g\|_{\infty} \leq nA \) under certain conditions.

Understanding the max-norm is crucial because it appears frequently in algebraic manipulations, especially when bounding polynomial coefficients.

Coefficients Bounding

When dealing with polynomial factorization, it is often essential to bound the coefficients of resulting polynomials. Coefficient bounding refers to setting limits on the absolute values of these coefficients, which provides insights into the polynomial's behavior and guides further manipulations.

• Bounding Techniques: We use inequalities to establish a range or maximum value the coefficients may reach. In the exercise, for \(|u| = |v|\), we derive \(|g_{i}| \leq \frac{(i+1)A}{|v|}\).
• Parameters Impact: The factors \(u\), \(v\), and the degree affect the bounds. Different relationships between \(u\) and \(v\)—such as when \(|u/v| < 1\) result in different bounds due to geometric series properties, leading to telescoped terms.
• Application: Coefficient bounding ensures the accuracy of algebraic processes and checks polynomial stability, particularly when solving equations or considering higher-order derivatives.

These bounds are not just numbers but powerful tools for analyzing polynomial equations and ensuring their manipulations remain within expected ranges.

Inequalities in Algebra

Inequalities play a pivotal role in algebra, serving as the foundation for understanding limits and behaviors of functions and expressions. They provide mathematical ways to express that certain conditions hold for values of a variable or series of numbers.

• Basic Principles: They allow us to compare two expressions with symbols like \(<, \leq, >, \geq\), thus establishing a range or constraint.
• Use in Algebra: In polynomial mathematics, inequalities help establish bounds for coefficients and resultant values, as seen in bounding exercises for functions like the given polynomial factorization problem.
• Advanced Applications: They extend to optimization problems, limit calculations, and differential equations, illustrating when certain solutions or transformations are possible or constrained.

Inequalities not only allow us to frame problems but also help generate solutions by narrowing the scope of possibilities, thus ensuring accurate and consistent results. Understanding inequalities is essential for proficiency in algebra and beyond.