Question

Let \(F\) be a field supporting the FFT, and \(a, b, q, r \in F[x]\) such that \(a=q b+r\) and deg \(r<\) \(\operatorname{deg} b \leq \operatorname{deg} a

Short answer

Perform three FFTs; compute \(q\) if the remainder is zero.

Step-by-step solution

Recognize the Problem

We're given polynomials \(a, b, q,\) and \(r\) from a field \(F[x]\) supporting the Fast Fourier Transform (FFT), under the condition \(a = qb + r\) with \(\deg r <\deg b \leq \deg a < n\). Our task is to decide whether \(r = 0\) and, if so, compute \(q\). This problem uses polynomial division and properties of the FFT.

Prepare for Fast Fourier Transform

To use the FFT, first ensure \(b\) is coprime to \(x^n - 1\). Begin by expressing both \(a\) and \(b\) as coefficient lists of length \(n\) by zero-padding if necessary, since \(b\)'s degree is less than \(n\).

Perform Multiplicative Inversion

Compute the multiplicative inverse of \(b\) modulo \(x^n - 1\) using the extended Euclidean algorithm. This inverse allows division in the Fourier Transformed domain.

Apply FFT to Polynomials

Compute the FFT of \(a\), \(b\), and the inverse of \(b\). These transforms convert polynomial multiplication and division into element-wise operations in the frequency domain.

Compute Quotient in Frequency Domain

Perform element-wise multiplication of the FFT of \(a\) with the FFT of the inverse of \(b\). This gives the FFT of the quotient \(q\).

Determine Remainder

Compute the inverse FFT of the product from Step 5. The resulting polynomial should be zero (or very close to zero) if \(r = 0\).

Decision Making

Check whether the resulting polynomial from Step 6 is the zero polynomial up to numerical precision. If it is, then \(r = 0\).

Compute the Final Quotient

If \(r = 0\), the quotient \(q\) is already computed as the inverse FFT from Step 6; otherwise, \(q\) is undefined.

Key concepts

Polynomial Division

Polynomial division is fundamental in algebra, especially when working with polynomials over a field. In this context, we are given two polynomials, \(a\) and \(b\), and need to determine if a remainder \(r\) is zero after division. The division process can be likened to numerical long division. Here, we aim to express the dividend \(a\) as a product of the quotient \(q\) and the divisor \(b\), with remainder \(r\). The equation follows the form \(a = qb + r\), where \(\deg r < \deg b\).

• Ensure that polynomials fit within a specified degree for efficient processing.
• Use zero-padding if needed to standardize polynomial length, crucial for applying FFT.

The challenge is computing this division efficiently, especially in algebraic tasks like solving polynomial equations. When approached correctly, polynomial division simplifies problems, allowing further operations like multiplicative inversion.

Multiplicative Inversion

Multiplicative inversion in the context of a field is the process of finding a number which, when multiplied with a given number, results in the multiplicative identity, which is 1. When dealing with polynomials, specifically in algebra, finding the inverse requires more intricate steps. For a polynomial \(b\) in a field \(F[x]\), the goal is to find another polynomial \(b^{-1}\) such that \(b \cdot b^{-1} \equiv 1 \mod (x^n - 1)\).

• Required for division operations in polynomial rings, especially important in FFT-related division.
• Utilizes algorithms like the Extended Euclidean Algorithm to solve for the inverse.

This step is crucial because it converts division in the polynomial domain into multiplication in the transformed Fourier domain, drastically simplifying the computation steps that follow.

Euclidean Algorithm

The Euclidean algorithm is a classical method for finding the greatest common divisor (GCD) of two numbers, and it adapts beautifully to polynomials. In this exercise, it's crucial for computing the multiplicative inverse of a polynomial. Extended versions of this algorithm enable us also to express the GCD as a linear combination of the two numbers.

• The classic Euclidean algorithm is iterative, subtracting multiples of the divisor from the dividend.
• In its extended form, it provides comprehensive results including coefficients for the linear combination.
• This extended form is essential for solving equations and finding inverses in polynomial rings.

By leveraging the Euclidean algorithm, we effectively prepare our polynomials for division in a field that supports FFT, establishing a path to finding the necessary multiplicative inverses, which are key for further polynomial operations.

Field Theory

Field theory is a vital area of abstract algebra that studies algebraic structures known as fields. A field is a set equipped with two operations, addition and multiplication, that satisfy specific properties like associativity, commutativity, and distributivity. Additionally, every non-zero element must possess a multiplicative inverse.

• Fields are crucial for understanding algebraic operations involving elements like numbers or polynomials.
• In this context, the polynomial field \(F[x]\) provides a framework for our computations, allowing us to utilize FFT on polynomials.
• Operations within a field include polynomial division, essential for the step-by-step solution.

Understanding the underlying field ensures we approach polynomial problems with effective and accurate methods, maintaining the properties required for mathematical consistency, especially when implementing FFT techniques.