Question

Let \(F\) be a field of characteristic zero and \(r, s, t, U \in F[x]\) such that \(r \cdot E U-s \cdot U=t\) and \(\operatorname{gcd}(r, s)=1\). Suppose that \(\operatorname{deg} g c d(r, t) \geq 1\), and reduce this difference equation for \(U\) to a difference equation for a proper divisor \(U^*\) of \(U\), with coefficients \(r^*, s^*, t^*\) of degrees no larger than those of \(r, s, t\). Answer the same question when deg \(\operatorname{gcd}(s, t) \geq 1\).

Short answer

Reduce the equation using the gcd of \(r, t\) and \(s, t\) to express \(U = d \cdot U^*\).

Step-by-step solution

Understand the Problem

We have the equation \(r \cdot EU - s \cdot U = t\) and the condition \(\operatorname{gcd}(r, s) = 1\). We need to reduce this equation under two separate cases: \(\operatorname{deg} \gcd(r, t) \geq 1\) and \(\operatorname{deg} \gcd(s, t) \geq 1\).

Case 1: gcd(r, t)

When \(\operatorname{deg} \gcd(r, t) \geq 1\), let \(d = \gcd(r, t)\). We can express \(r = d \cdot r_1\) and \(t = d \cdot t_1\). Substitute these into the original equation to get \(d \cdot r_1 \cdot EU - s \cdot U = d \cdot t_1\). Factor out \(d\) to reduce this to \(r_1 \cdot EU - \frac{s}{d} \cdot U = t_1\).

Proper Divisor Adjustment for Case 1

Since \(d\) divides both \(r\) and \(t\) but not \(s\), it naturally divides \(U\), allowing us to set \(U = d \cdot U^*\). This adjustment leads to the equation \(r_1 \cdot EU^* - \frac{s}{d} \cdot U^* = t_1\) for the smaller polynomial \(U^*\).

Case 2: gcd(s, t)

For \(\operatorname{deg} \gcd(s, t) \geq 1\), let \(d = \gcd(s, t)\). Then express \(s = d \cdot s_1\) and \(t = d \cdot t_1\). Substituting these into the original equation yields \(r \cdot EU - d \cdot s_1 \cdot U = d \cdot t_1\). Factor out \(d\) to give \(r \cdot EU - s_1 \cdot U = t_1\).

Proper Divisor Adjustment for Case 2

Since \(d\) divides both \(s\) and \(t\), it also divides \(U\), allowing the reduction to \(U = d \cdot U^*\). The equation simplifies to \(r \cdot EU^* - s_1 \cdot U^* = t_1\).

Key concepts

Field of Characteristic Zero

A field of characteristic zero is an important algebraic structure used to study polynomial equations.
Characteristics of a field describe whether or not sums of multiple copies of the number one, i.e., 1 + 1 + ... + 1, eventually equal zero.
For a field with characteristic zero, this never happens, meaning the field behaves like the set of rational numbers in terms of addition.
A common example is the field of real numbers,

• where no matter how many copies of 1 you add together, it will not sum to zero unless you involve subtraction of equal value.
• The characteristic zero property is crucial for maintaining certain algebraic rules when dealing with polynomials.

This property ensures that you can perform algebraic manipulations without encountering unexpected results that would occur in fields with a different characteristic.

GCD (Greatest Common Divisor)

The greatest common divisor (GCD) of two numbers or polynomials is the largest polynomial that divides both without leaving a remainder.
In the context of polynomials, finding the GCD is crucial in simplifying expressions and equations.
When working with the equation in the given exercise, knowing that \(\operatorname{gcd}(r, s) = 1\) means that \(r\) and \(s\) are coprime polynomials.

• In other words, they have no common factors except for constants, which allows certain simplifications.
• The degree of the GCD, when greater than or equal to one, indicates that there is a non-trivial polynomial that divides both polynomials.

By utilizing the GCD of \(r\) or \(s\) with \(t\), the problem simplifies since it allows us to reduce the original equation to an equivalent one involving smaller polynomials.

Polynomial Degrees

Polynomial degrees are fundamental in algebra as they indicate the highest power of the variable in a given polynomial.
For example, a polynomial \( p(x) = x^3 + 2x^2 - 4 \) has a degree of 3 because the highest power of \(x\) is 3.
The degree is important for understanding the behavior of polynomials, especially within equations where balancing these degrees is key.

• In the exercise, reducing the polynomial \(U\) into a new polynomial \(U^*\) without increasing the degrees of \(r, s,\) and \(t\) is essential.
• This ensures that the complexity of the equation does not increase unnecessarily, making solutions more manageable.

Thus, working within the constraints of polynomial degrees often leads to insights on solving polynomial equations more effectively.

Difference Equation

A difference equation is a key concept in algebra, where it describes relationships between values at successive points, typically involving sequences or polynomials.
In simpler terms, it is an equation that expresses the difference between terms in a sequence.
In the context of the exercise, the given equation is a form of a difference equation for polynomial sequences.

• When working through reduction, as seen in the solution steps, a difference equation allows breaking down the expression for \(U\) into simplified parts.
• Usually, you look for conditions or transformations that allow for these simplifications, such as finding a GCD or factoring terms.

In many applications, this helps in predicting future values in a sequence or finding simplified forms of complex algebraic expressions efficiently.