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

Use FFTs to transform, divide and inverse transform to decide if \( r=0 \) and find \( q \).

Step-by-step solution

Understand the Problem

You need to check if the remainder \( r \) is zero when dividing a polynomial \( a \) by another polynomial \( b \), using properties of FFT (Fast Fourier Transform). If \( r = 0 \), you also need to find the quotient \( q \). This is possible because \( b \) and \( x^n - 1 \) are coprime and the polynomials are in a field that supports FFT.

Initial Setup with FFT

Identify the degree \( n \), a power of 2, such that all necessary polynomials can be embedded within it for the FFT. The input polynomials \( a \) and \( b \) should both be represented using \( n \) coefficients by padding with zeros as necessary. The goal is to use FFT to efficiently perform polynomial operations.

Compute FFTs of Polynomials

Compute the FFTs of \( a \) and \( b \). This will transform the polynomials from their coefficient representation into their point-value representation. Denote these transformed polynomials as \( A \) and \( B \) respectively.

Perform Pointwise Division

For each point obtained from the FFT, divide the corresponding value of \( A \) by the value of \( B \). This yields a new point-value representation which represents the division of the polynomials in the frequency domain. Note that division is possible because \( b \) is coprime to \( x^n - 1 \) and hence none of the values \( B(i) \) will be zero.

Inverse FFT to Obtain Coefficients

Perform the inverse FFT on the quotient from the previous step to transform it back from point-value form into coefficient form. This yields the coefficients of the quotient polynomial \( q \).

Check Remainder Coefficients

After obtaining \( q \), multiply it by \( b \) and subtract from \( a \) to check if \( r = a - q \cdot b = 0 \). If all coefficients of \( r \) are zero, then the remainder is zero and you have found the quotient \( q \).

Verification and Conclusion

Verify the above computations with a recomputation if necessary. If \( r = 0 \), then \( q \) has been successfully computed using three applications of FFT: transform, division, and inverse transform in the FFT-supported field.

Key concepts

Polynomial Division

Polynomial division is a process to divide one polynomial by another, much like dividing numbers. In a field that supports the Fast Fourier Transform (FFT), this operation can be performed efficiently. Let's break it down further:

1. **Identifying Degrees**: When dividing a polynomial \( a \) by a polynomial \( b \), it's important to keep track of their degrees. The remainder \( r \) must always have a degree less than \( b \). This condition simplifies the operation, as it establishes a clear boundary for the remainder. 2. **FFT Setup**: With FFT, the polynomials \( a \) and \( b \) are padded with zeros to fit within a degree \( n \), which is a power of two. This setup is crucial as FFT algorithms work most efficiently with input sizes that are powers of two.3. **Transformation**: Instead of working with complex coefficient representations, FFT transforms the polynomial data into point-value representations. This transformation allows polynomial operations to be conducted with simple arithmetic.

Using polynomial division through FFT, we achieve a significant reduction in computational complexity, making it a powerful tool for solving polynomials within an FFT-supported field.

Remainder Theorem

The Remainder Theorem is a useful tool in polynomial algebra. It helps us determine the remainder when dividing one polynomial by \( x - c \). Let's delve into how it applies to the FFT context:

- **Basic Idea**: The theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is simply \( f(c) \). This principle extends to our problem where we need to check if a remainder \( r \) is zero.- **Checking Zeros with FFT**: After calculating the quotient \( q \) and remainder \( r \), we verify if \( r = 0 \). In practice, this means checking the coefficients of \( r \) obtained after subtracting \( q \cdot b \) from \( a \). If all these coefficients are zero, then indeed, \( r = 0 \).

This theorem simplifies the verification step in polynomial division by providing a straightforward condition to check for a zero remainder, leveraging both algebraic properties and FFT efficiencies.

Coprime Polynomials

Coprime polynomials are polynomials with no common factors other than a constant. Understanding this concept is crucial in division problems involving the Fast Fourier Transform:- **Coprime Condition**: In our exercise, the polynomial \( b \) needs to be coprime to \( x^n - 1 \). Why is this important? It ensures that during the FFT transformation, no zeros occur in \( B(i) \). This allows division to occur smoothly, as each point from \( A \) can be divided by corresponding points from \( B \).- **Ensuring Smooth Division**: Since \( b \) and \( x^n - 1 \) share no non-trivial common factors, for every Fourier point, the value of \( b \) remains non-zero. This non-zero condition is essential, as any zero value would lead to undefined behavior (division by zero) during the point-wise division stage.Utilizing coprime properties in polynomial division ensures the solution's validity, particularly when employing FFT, by guaranteeing a consistent divide step across transformed point-value pairs.