Question

Compute a square root \(g \in \mathbb{Q}[x]\) of \(f=1+4 x \in \mathbb{Q}[x]\) modulo \(x^{8}\) such that \(g(0)=1\), using Newton iteration.

Short answer

The square root of \(f = 1 + 4x\) is \(g = 1 + 2x - 2x^2\) modulo \(x^8\).

Step-by-step solution

Define Initial Guess

We start with an initial guess for the square root of \(f\) modulo \(x^8\). Given \(g(0) = 1\), we can use \(g_0 = 1\) as our initial guess. This satisfies the condition at \(x = 0\).

Apply Newton's Iteration Step

Newton's method provides a way to improve the approximation iteratively. Given our function \(f = 1 + 4x\) and the initial guess \(g_0\), the formula for the next iteration of Newton's method for square roots is: \[ g_{n+1} = g_n - \frac{g_n^2 - f}{2g_n} \].

Simplify the Newton Formula

Since we're working in \(\mathbb{Q}[x]/(x^8)\), the iteration formula becomes: \[ g_{n+1} = g_n - \frac{g_n^2 - (1 + 4x)}{2g_n} \]. Because \(g_n = 1\) initially, we compute: \[ g_1 = 1 - \frac{1^2 - (1 + 4x)}{2 \times 1} = 1 + 2x \].

Iterate to Higher Orders

Continuing from \(g_1 = 1 + 2x\), we compute the next approximation: \[ g_2 = g_1 - \frac{g_1^2 - (1 + 4x)}{2g_1} \]. First, compute \(g_1^2 = (1 + 2x)^2 = 1 + 4x + 4x^2\), thus \[ g_2 = (1 + 2x) - \frac{1 + 4x + 4x^2 - (1 + 4x)}{2(1 + 2x)} = 1 + 2x - \frac{4x^2}{2 + 4x} \].

Further Simplification

For \(g_2\), simplify as: \[ g_2 = (1 + 2x) - 2x^2 = 1 + 2x - 2x^2 \], reducing modulo \(x^8\). Repeat this Newton iteration until \(g_n^2 \equiv 1 + 4x \pmod{x^8}\).

Finalize Solution

Upon iterating the Newton step enough times, reach a solution that satisfies the initial condition \(g(0)=1\) and is accurate modulo \(x^8\). The representation will not have higher-order terms beyond \(x^7\), as computations are reduced modulo \(x^8\).

Key concepts

Square Root Computation in Polynomials

Understanding how to find the square root of a polynomial involves a fascinating process. Here, we aim to compute a square root of the polynomial function \( f = 1 + 4x \) under modulo \( x^8 \). This simply means that we are looking for another polynomial \( g \) such that \( g^2 \approx f \) but only up to terms with power less than 8.

To break this down, consider the analogy of finding a square root of a decimal number but limited to a certain decimal place accuracy. Similarly, in the realm of polynomials, we are interested in finding accurate representations only up to the specified polynomial degree. The condition \( g(0) = 1 \) adds an additional constraint, directing us to initiate the solution by setting \( g(0) \) to start at constant value 1. This serves as the trivial square root since at zero, the polynomial function is simply a constant.

Overall, the goal in computing polynomial square roots this way is to find a polynomial function that fits when squared, adhering to constraints imposed by the polynomial degree cut-off. This approach holds particular utility in algebraic manipulations and solving higher-order polynomial equations with precision.

Polynomial Approximation

Polynomial approximation is a fundamental concept in numerical analysis. It involves representing a complex function more simply, usually to simplify calculations. In our context, approximating \( f = 1 + 4x \) involves finding another polynomial \( g \) such that \( g^2 \) closely reflects \( f \).

When approximating polynomials, one must consider the degree or order of approximation, as this determines which terms of the polynomial are considered relevant. Here, we modulate all computations by \( x^8 \), meaning our focus is only on finding a precise match up to the 7th power of \( x \). Essentially, higher-degree terms are ignored because their influence is negligible or outside the required degree of precision.

By iteration through Newton's method, our objective is akin to peeling back layers of an onion: each step provides a closer approximation where complexity is incrementally added with higher-degree terms. This iterative process ensures that \( g_n \), our evolving solution, becomes ever closer to the true square root of \( f \).

Through Newton's method, polynomial approximation proves to be a powerful tool, effectively refining solutions and enabling advanced problem-solving in polynomial algebra.

Newton Iteration Process

The Newton iteration process is a technique widely applied to refine approximations, here harnessed to find square roots of polynomials. This process begins with an initial guess — in our case, \( g_0 = 1 \), and iteratively refines this guess until a satisfactory precision is reached.

The central idea of Newton's method is to utilize a tangent approach: starting from this initial base, the next approximation is determined using the formula \[ g_{n+1} = g_n - \frac{g_n^2 - f}{2g_n} \]. This essentially means we iteratively adjust our guess based on the error from the desired outcome, smoothing disparities with each cycle.

• The numerator \( g_n^2 - f \) represents the deviation of our current guess squared from the target polynomial \( f \).
• The denominator \( 2g_n \) helps scale the correction, finely tuning \( g \) towards \( \sqrt{f} \).

Each iteration, \( g_1, g_2, \ldots \), offers improved accuracy, driving \( g \) closer to being an accurate representation of the square root function. It's worth noting that the success of this method largely hinges on suitable initial guesses and the iterative reduction of errors through smart division and subtraction in calculations.

Newton's iteration is used broadly across mathematical fields to zero in on solutions, essential particularly in polynomial computations where precision is non-negotiable.