Question

Use Newton iteration to compute \(f^{-1} \bmod x^{5}\) for \(f=x^{2}-2 x+1 \in \mathbb{Q}[x]\).

Short answer

The inverse modulo is \( 4x - 6x^2 + 4x^3 - 4x^4 \).

Step-by-step solution

Function and Derivative

First identify the function and its derivative. We are given the function \( f(x) = x^2 - 2x + 1 \). The derivative of this function is calculated as \( f'(x) = 2x - 2 \).

Initial Inverse Guess

Next, we make an initial guess for the inverse modulo. A common choice for polynomials near \(x\) is to start with the inverse around a simple leading term, such as \(g_0 = 1\).

Newton's Iteration Formula

Apply Newton's iteration formula for inverse functions: \( g_{n+1} = 2g_n - g_n^2 f \) being reduced modulo \( x^m \). Here, we are given \( m = 5 \).

First Iteration

Calculate \( g_1 \) by using the formula: \[g_1 = 2g_0 - g_0^2 f \]Substitute \( g_0 = 1 \) and \( f = x^2 - 2x + 1 \):\[g_1 = 2 \times 1 - 1^2(x^2 - 2x + 1) = 2 - (x^2 - 2x + 1) = 1 - x^2 + 2x\]Therefore, \( g_1 = 2x - x^2 \).

Second Iteration

Perform another Newton's iteration:\[g_2 = 2g_1 - g_1^2 f \]Substitute \( g_1 = 2x - x^2 \) and simplify further.First, calculate \( g_1^2 f \):\[g_1^2 = (2x - x^2)^2 = 4x^2 - 4x^3 + x^4\]Multiply by \(f = x^2 - 2x + 1\):\[g_1^2 f = (4x^2 - 4x^3 + x^4)(x^2 - 2x + 1) = 4x^4 - 8x^5 + 4x^2 - 4x^3 + x^4\]It is only necessary to compute and use terms up to \(x^4\):\[g_2 = 2(2x - x^2) - (4x^4 - 4x^3 + 4x^2) = 4x - 2x^2 - 4x^4 + 4x^3 - 4x^2\]Collect and simplify:\[g_2 = 4x - 6x^2 + 4x^3 - 4x^4\]

Conclusion

The polynomial resulting from Newton's iteration gives the inverse modulo. Therefore, the inverse of \(f\) modulo \(x^5\) is:\[g(x) = 4x - 6x^2 + 4x^3 - 4x^4\]

Key concepts

Polynomial Inversion

Polynomial inversion is an exciting concept and often relies on ingenious methods like Newton's iteration for calculating the inverse of a polynomial modulo another polynomial.To better understand, think of inversion like reversing a process. If you can solve the equation for one process, inversion lets you go the other way.

The aim in polynomial inversion, especially in a modular sense, is to find a polynomial that is the multiplicative inverse of the original polynomial under a given modulus.For our particular exercise, we set out to determine the inverse of polynomial \( f(x) = x^2 - 2x + 1 \) modulo \( x^5 \).

• We calculate the inverse using steps that include guessing an initial inverse, sometimes making assumptions based on leading terms.
• The process involves careful applications of mathematical formulas to reduce error and refine the inversion.

Polynomial inversion is not only about flipping a function but becoming adept at managing calculations and ensuring that operations stay within certain bounds, like those set by a modulus.

Derivative Computation

Derivative computation is foundational for understanding further steps in mathematical algorithms including polynomial inversion.

In the context of this exercise, identifying the derivative of the given polynomial \( f(x) = x^2 - 2x + 1 \) is a critical step. The derivative, in this case, computed as \( f'(x) = 2x - 2 \), forms the basis for Newton's iteration.Newton's iteration relies on both the function and its derivative.

• Computation of derivatives lets us understand the slope or rate of change of a polynomial at any point which is crucial for iteration processes.
• Accurate determination of the derivative allows us to better apply numerical methods to hone down on the inverse we seek.

The derivative tells us how a small change in input affects output, making it an indispensable tool.

Modular Arithmetic

Modular arithmetic is like clock arithmetic, where numbers wrap around once they reach a certain value—the modulus.

In our exercise, modular arithmetic helps contain polynomial operations within the bounds determined by the modulus \( x^5 \). Thus, we only consider terms up to \( x^4 \), effectively managing the scope of polynomial terms during operations.

• It ensures that processes do not exceed the defined polynomial degree, keeping calculations efficient and manageable.
• Being modular also means the solution remains valid within those bounds, consistently delivering expected results even when reducing higher degree terms.

Understanding modular arithmetic is not just about managing complexity but about holding the integrity of our operation within a defined system. It permeates through various aspects, including handling polynomials efficiently, making it critical in algorithmic contexts like the one described.