Question

Compute the gcd of \(x^{4}-3 a x^{3}-8 a^{2} x^{2}+18 a^{3} x-8 a^{4}\) and \(x^{3}-\) \(a x^{2}-8 a^{2} x+6 a^{3}\) in \(\mathbb{Q}[x, a]\) using the small primes modular algorithm.

Short answer

The gcd is \( x^2 + 2ax - 4a^2 \).

Step-by-step solution

Understand the Polynomials

Identify the polynomials given in the exercise. Let's call them:\[ P(x) = x^4 - 3ax^3 - 8a^2x^2 + 18a^3x - 8a^4 \]\[ Q(x) = x^3 - ax^2 - 8a^2x + 6a^3 \]Both polynomials are in \( \mathbb{Q}[x, a] \), meaning they have coefficients in the rational numbers.

Select Small Prime Moduli

To use the small primes modular algorithm, select small prime numbers to reduce the problem into smaller computations. Possible small primes to choose are 2, 3, and 5.

Compute GCD under Modulo for Small Primes

For each selected small prime, reduce the coefficients of the polynomials modulo that prime. Then, compute the gcd of the resulting polynomials over the integers \( \mathbb{Z}/p\mathbb{Z}[x] \).- For example, if \( p = 2 \), compute the gcd of \( P(x) \mod 2 \) and \( Q(x) \mod 2 \).

Lift the GCD to Integer Coefficients

After finding the gcd for each modulo \( p \), use the results to determine the gcd over the original field of rational coefficients using the Chinese Remainder Theorem or interpolation techniques.

Verify the Results

To ensure correctness, verify that the computed gcd divides both original polynomials. Also, check consistency among different small primes used in computations.

Key concepts

Small Primes Modular Algorithm

The small primes modular algorithm is a practical approach to simplify polynomial computations by reducing the problem to several smaller, more manageable tasks. The core idea is to compute the gcd of polynomials by first considering their equivalent forms under various small prime numbers. Essentially, when you work with polynomials having rational coefficients, computations can become quite complex.

• Select small prime numbers, such as 2, 3, or 5, which will be used to reduce polynomial coefficients modulo these primes.
• This reduction simplifies the polynomials into forms that are easier to handle and operate with. With these simpler forms, you compute the gcd for each polynomial pair.
• By solving the gcd under these modular constraints, you break down the complex problem into smaller pieces that are much easier to solve.

Once the gcd is found under these small primes, the problem can then be reconstructed into its original form through other mathematical techniques, ensuring accuracy and consistency.

Polynomial Coefficients in Rational Numbers

Polynomial coefficients in rational numbers mean that each part of the polynomial is a fraction, of the form \( \frac{a}{b} \), where both \( a \) and \( b \) are integers. This can complicate computations because rational number arithmetic is generally more involved than simple integer arithmetic.
To manage this, when computing a gcd, it's helpful to convert these rational coefficients into approximations using modular arithmetic:

• Firstly, transform the rational polynomial coefficients into integers by using modular reductions, which drastically simplifies calculations.
• This allows you to calculate gcd using methods similar to integer polynomials.
• It is important to carefully handle these reductions to ensure no loss of important information regarding the gcd.

After working with reduced forms, the polynomial should be converted back, ensuring the gcd reflects correctly over the original rational coefficients.

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a powerful tool in number theory, especially when dealing with computational problems involving modular arithmetic. It essentially allows you to reconstruct the original problem by considering the results under multiple moduli.
Here's how it applies to polynomial gcd computations:

• Once you've determined the gcd of polynomials under several small primes, you might wonder how these individual results relate to the original problem.
• The CRT offers a way to combine these modular gcd results into a single result for the original problem in rational numbers.
• It works by finding a common number that satisfies the congruences from each small prime gcd result.

By strategically applying the CRT, you are assured that the gcd you derive using small primes is consistent with the polynomial over the rational numbers.

Interpolation Techniques

Interpolation techniques serve as a mathematical strategy to reconstruct a function, like polynomials, given a set of points provided by the smaller modular computations. In the context of gcd calculation, interpolation helps ensure our derived gcd is accurate for the original polynomial set.
When you've calculated gcd over small primes:

• Interpolation allows you to align these results with the polynomial's original domain.
• By doing this, you accurately reconstruct the gcd polynomial as it should appear in the rational number domain.
• This involves determining a unique polynomial that matches the modular gcd results at chosen points.

Overall, interpolation is a crucial final step in verifying and converting that the small prime solutions are correct when integrated back into the broader context of rational polynomial computations.