Question

Design and analyze an efficient algorithm for the following problem. The input is a pair of polynomials \(g, h \in \mathbb{Z}[X],\) along with their greatest common divisor \(d\) in the ring \(\mathbb{Q}[X] .\) The output is the greatest common divisor of \(g\) and \(h\) in the ring \(\mathbb{Z}[X]\)

Short answer

Question: Given polynomials g, h in the ring 𝕍[X], and their GCD in the ring 𝕋[X], describe an efficient algorithm for finding the GCD in the ring 𝕍[X] and analyze its efficiency. Answer: To find the GCD of the polynomials g, h in 𝕍[X], you can follow these steps: (1) Calculate the content of the polynomials, (2) Divide the given polynomials by their content, (3) Find the GCD of the rational polynomials, (4) Calculate the content of the GCD of the rational polynomials, (5) Divide the GCD by its content, and (6) Find the GCD in 𝕍[X] by multiplying the primitive GCD by the GCD of the content of the given polynomials. This algorithm is efficient since it uses the Euclidean algorithm for integers and the given GCD of the rational polynomials.

Step-by-step solution

Understand Polynomials and GCD

A polynomial is an expression consisting of variables and coefficients, with only non-negative integer exponents of variables. The Greatest Common Divisor (GCD) is the highest degree polynomial that divides two given polynomials without leaving a remainder. In this problem, we are given the polynomials \(g, h \in \mathbb{Z}[X]\), which means the coefficients of the polynomials are integers. We are also given their GCD in the ring \(\mathbb{Q}[X]\), which means the coefficients of the GCD are rational numbers. We need to find their GCD in the ring \(\mathbb{Z}[X]\) (with integer coefficients).

Calculate the content of the polynomials

The content of a polynomial is the GCD of its coefficients. We will first find the content (\(c(g)\) and \(c(h)\)) of the given polynomials \(g\) and \(h\). This can be done using the Euclidean algorithm for integers.

Divide the given polynomials by their content

Next, we will divide the given polynomials by their content. This will give us new polynomials \(\tilde{g}\) and \(\tilde{h}\) that have integer coefficients and are primitive (their content is 1). \(\tilde{g} = \frac{g}{c(g)}\), \(\tilde{h} = \frac{h}{c(h)}\)

Find the GCD of the rational polynomials

Now, we will find the GCD of the rational polynomials \(d \in \mathbb{Q}[X]\), which is already given in the problem statement.

Calculate the content of the GCD of the rational polynomials

In this step, we will find the content of the GCD of the rational polynomials (\(c(d)\)). This can be done using the Euclidean algorithm for integers.

Divide the GCD by its content

Divide the GCD of the rational polynomials by its content to obtain the GCD of the integer polynomials: \(\tilde{d} = \frac{d}{c(d)}\)

Find the GCD in \(\mathbb{Z}[X]\)

Finally, in order to get the GCD in the ring \(\mathbb{Z}[X]\), we need to multiply the primitive GCD \(\tilde{d}\) by the GCD of \(c(g)\) and \(c(h)\). \(\text{GCD}(g, h)_{\mathbb{Z}[X]} = \text{GCD}(c(g), c(h)) \cdot \tilde{d}\) This algorithm is efficient since it involves calculating the GCD of integer coefficients, which can be carried out using the Euclidean algorithm for integers, and uses the given GCD of the rational polynomials. The main computational task involves calculating the primitive GCD \(\tilde{d}\), which is derived efficiently from the known GCD in \(\mathbb{Q}[X]\).

Key concepts

Euclidean Algorithm

The Euclidean algorithm is an ancient and efficient method for finding the greatest common divisor (GCD) of two numbers. This extends beautifully to polynomials with integer coefficients as well. It operates on a simple principle: when you divide one number by another, the GCD of these two numbers is the same as the GCD of the divisor and the remainder. Therefore, we can 'step down' repeatedly using division until we reach a remainder of zero, at which point the last non-zero remainder is, in fact, the GCD.

Application to Polynomials

For polynomials, the Euclidean algorithm is slightly adapted. We divide the higher degree polynomial by the lower degree one and look at the remainder. This process is iterated, reducing the degree at each step, until we reach a remainder of zero. The last non-zero remainder is the GCD of the polynomials.

While the algorithm is simple, it's important to understand that it doesn't just work for numbers, but for any context where division with remainder makes sense, including the ring of polynomials with integer coefficients.

Rational Polynomials

Rational polynomials are polynomials whose coefficients are rational numbers, not just integers. When dealing with the GCD of such polynomials, interesting issues arise because the same polynomial can be written in different ways by multiplying or dividing through by a non-zero rational number.

Normalization

To standardize and simplify the process of finding the GCD of rational polynomials, it is a practice to focus on their 'normalized' or 'content-removed' forms — essentially turning them into their primitive polynomial counterparts. A 'primitive polynomial' is one whose coefficients have no common divisor besides 1 or -1, making them much easier to work with in GCD calculations.

Integer Coefficients

Polynomials with integer coefficients are ones in which every term's coefficient is a whole number. These polynomials belong to a ring denoted as \(\mathbb{Z}[X]\), where \(\mathbb{Z}\) refers to the set of integers and \(X\) is the variable of the polynomial. This constraint on the coefficients being integers is crucial when the polynomial is used in different algorithms, like the Euclidean algorithm.

When we have a polynomial with rational coefficients, as given in our problem, and we wish to find the GCD in \(\mathbb{Z}[X]\), we must consider the conversion of this polynomial into one with integer coefficients while retaining the property that reflects the greatest common divisor.

Polynomial Content

The concept of 'content' for a polynomial is analogous to the GCD for integers. Specifically, the content is the GCD of the polynomial's coefficients. Understanding content is critical when dealing with polynomials over the integers, as it helps us transition between polynomials with rational coefficients and polynomials that are 'primitive'.

Why Content Matters

Calculating the content allows us to find equivalent primitive polynomials, which are crucial in simplifying the computation of polynomial GCDs. The content can be stripped from the original polynomial by dividing each term by the content, yielding a polynomial with integer coefficients and greatly simplifying the task of finding the GCD within \(\mathbb{Z}[X]\).