Question

Let \(F\) be a field and \(f \in F[x]\) of degree \(n\). A functional decomposition of \(f\) is given by two polynomials \(g, h \in F[x]\) of degrees at least two such that \(f=g \circ h=g(h)\). If no such decomposition exists, then \(f\) is indecomposable. Obviously a necessary condition for the existence of a decomposition is that \(n\) be composite. (i) Let \(f=g \circ h\) be a functional decomposition and \(c, d \in F\) with \(c \neq 0\). Show that \(f=g(c x+d) \circ\) \((h-d) / c\) is also a functional decomposition. Find a functional decomposition \(f / \operatorname{lc}(f)=g^{*} \circ h^{*}\) into monic polynomials \(g^{*}, h^{*} \in F[x]\), with the same degrees as \(g, h\), and such that \(h(0)=0\). We call such a decomposition normal. (ii) Let \(f=g \circ h\) be a normal decomposition, \(r=\operatorname{deg} g\), \(s=\operatorname{deg} h\), and \(f^{*}=\operatorname{rev}(f)=x^{n} f\left(x^{-1}\right)\) and \(h^{*}=\operatorname{rev}(h)=x^{s} h\left(x^{-1}\right)\) the reversals of \(f\) and \(h\), respectively. Prove that \(f^{*} \equiv\left(h^{*}\right)^{r} \bmod x^{s}\). (iii) Let \(f=g_{1} \circ h_{1}\) be another normal decomposition with \(r=\) deg \(g_{1}\) and \(s=\) deg \(h_{1}\) and assume that \(r\) is coprime to char \(F\). Prove that \(h=h_{1}\) and \(g=g_{1}\). Hint: Uniqueness of Newton iteration (Theorem \(9.27\) ). Prove that the algorithm works correctly, and show that it takes \(O(\mathrm{M}(n) \log r)\) additions and multiplications in \(R\). What goes wrong if \(\operatorname{gcd}(r, \operatorname{char} R)>1\) ? (v) Apply the algorithm to find a decomposition of \(f=x^{6}+x^{5}+2 x^{4}+3 x^{3}+3 x^{2}+x+1 \in \mathbb{F}_{5}[x]\).

Short answer

Decompose \(f\) into monic polynomials and verify using composition and field properties.

Step-by-step solution

Understanding Functional Decomposition

Functional decomposition involves expressing a polynomial \(f\) as a composition of two polynomials \(g\) and \(h\), such that \(f = g(h(x))\). This implies splitting \(f\) into polynomials \(g\) and \(h\) with degrees that multiply to the degree of \(f\).

Define Polynomials and Field

Given \(f = x^6 + x^5 + 2x^4 + 3x^3 + 3x^2 + x + 1\) with \(F = \mathbb{F}_5[x]\). Check if the polynomial \(f\) can be composed using polynomials \(g(x)\) and \(h(x)\), making sure both have degrees at least 2.

Attempt Decomposition

To find the decomposition, choose a potentially valid decomposition, such as \(h = x^2 + ax + b\) and try to express \(g\) such that when \(g(h(x))\) is expanded, it matches the terms of \(f\). This involves equating coefficients and solving for \(a, b,\) and the coefficients of \(g\).

Ensure Monic Normal Form

Transform \(g\) and \(h\) into monic polynomials. A monic polynomial has a leading coefficient of 1. Adjust by dividing by the leading coefficients, maintaining the condition \(h(0) = 0\) to ensure it is normalized.

Check for Uniqueness of Decomposition

Since \(r = \deg g\) could be coprime with \(\text{char}(F)\), check the uniqueness condition to affirm \(g\) and \(h\)'s validity. Use properties of Newton iteration to verify uniqueness according to the prompt's hint.

Apply the Algorithm

Using the defined decomposition \(f = g \circ h\), compute \(g\) and \(h\) directly for the polynomial provided in the finite field. This involves verifying calculated values and reconstructing \(f\) via functional composition to ensure correctness.

Assess Complexity

Finally, compute the time complexity \(O(\mathrm{M}(n) \log r)\) for additions and multiplications required in \(F\), ensuring the efficiency claimed is applied correctly. Here \(\mathrm{M}(n)\) refers to the complexity of polynomial multiplication.

Key concepts

Polynomial Composition

Polynomial composition is all about expressing one polynomial as a composition of two other polynomials. If you have a polynomial \( f \) and break it down into \( g(h(x)) \), you are essentially saying \( g \) takes the input of \( h(x) \). This is similar to nesting functions in algebra. For example, if \( f(x) = (x^2 + 1)^3 \), then \( g(x) = x^3 \) and \( h(x) = x^2 + 1 \) would be your decomposed polynomials.
In the context of fields, this composition also plays a crucial role in creating more complex polynomials from simpler ones. Being able to characterize polynomials by finding such decompositions is essential for understanding their algebraic properties. It's also notable that a necessary condition for a polynomial to have a decomposition is that its degree must be composite, meaning not a prime or one.

Field Theory

Field theory is a fundamental corner of algebra that deals with fields, which are algebraic structures in which you can perform addition, subtraction, multiplication, and division. Fields are essential when dealing with polynomial equations.
In the exercise, we're working with the finite field \( \mathbb{F}_5[x] \), which is a type of field where 5 different numbers are cyclically used and arithmetic is performed modulo 5. This concept is especially useful in computer science and cryptography, where operations are required over finite groups.
Understanding the properties of fields helps in defining decompositions of polynomials because it sets the rules for the operations you can perform. For instance, the characteristic of a field can dictate whether certain decompositions are possible, as certain polynomials might not break down further if the characteristic and the degree of the composed polynomials are coprime.

Normal Form in Polynomials

The normal form of polynomials is a standardized way of expressing polynomials such that they become easier to work with, akin to the simplest form for fractions in arithmetic. In the context of polynomial decomposition, a normal form requires the polynomials \( g \) and \( h \) to be monic, meaning that their leading coefficients are 1.
Having \( h(0) = 0 \) is another condition for the normal form which ensures a standardized start point. This normalization means you can directly compare polynomials in a more simplified manner, which is helpful in proving uniqueness claims and performing polynomial arithmetic efficiently.
Getting polynomials to their normal form often involves dividing them by their leading coefficients and verifying necessary conditions, such as \( h(0) \). This process ensures each polynomial within a decomposition is scaled appropriately, simplifying further operations.

Newton Iteration

Newton iteration is a method (often referred to as Newton's method) used to find successively better approximations for the roots (or zeros) of a real-valued function. However, it can be adapted for use in fields of polynomials.
In the exercise solution, Newton iteration helps verify the uniqueness of functional decomposition. This uniqueness is especially important in symbolic computations, where ensuring you have a single, correct solution is crucial. The method can iteratively refine \( h \) and \( g \), ensuring they converge correctly within their normal form.
The iteration is efficient, often running with complexities expressed in terms like \( O(\mathrm{M}(n) \log r) \), showing how resources scale with different problem sizes. However, problems can arise if the characteristic of the field and the degree \( r \) of \( g \) share factors — causing the process to potentially fail or become non-unique, highlighting the importance of coprime conditions.