Exploring Polynomial Solutions: A Deep Dive into Conditions for Polynomial Composition

In the realm of polynomial algebra, the problem of finding polynomials that satisfy specific conditions, particularly those involving compositions, can be both intriguing and challenging. This discussion delves into the process of identifying polynomials that satisfy certain compositional properties, specifically focusing on the quadratic polynomial x^2 - 2.

Introduction to Polynomial Compositions

Polynomials with integer coefficients are a fundamental topic in algebra. Consider a polynomial P with integer coefficients. When we plug in x sqrt{2}, we get the equality:

[ P(sqrt{2}) a text{ for some } a in mathbb{Z} ]

This implies that the polynomial x^2 - 2 divides P(x^2 - 2) - a. Through this, we can express P as a polynomial in x^2 - 2, leading to:

[ P^2 b_m x^2 - 2^{m} b_{m-1} x^2 - 2^{m-1} ldots b_0 text{ for some } b_i in mathbb{Q} ]

Notably, since P^2 is an even polynomial, P must either have only odd powers or only even powers of x.

Monoid Structure and Polynomial Composition

The set of polynomials with integer coefficients, denoted as mathbb{Z}[x], forms a monoid under polynomial composition. This means the operation of composition is associative and the identity element is the identity polynomial E(x) x.

Suppose P has degree 0. Then:

[ P(x) p text{ and } P text{ satisfies } p^2 - 2 p ]

This gives the solutions p 2 or p -1.

Polynomial Composition with Q(x) x^2 - 2

Now consider the polynomial Q(x) x^2 - 2. We aim to find polynomials P such that:

[ Q circ P P circ Q ]

The key observation here is that finite compositions of Q satisfy this equation. Therefore, we conclude:

[ P underbrace{Q circ Q circ ldots circ Q}_{m text{ times}} ]

Choosing m leq 1012 gives polynomials of the correct degree. This approach explicitly constructs a series of polynomials that satisfy the given condition.

Linear Combinations and Polynomial Maps

One might wonder about the role of linear combinations. However, they do not generally satisfy the equation in question. Instead, we can represent every even polynomial in the basis:

1 Q Q circ Q Q circ Q circ Q …

and every odd polynomial in the basis:

x x cdot Q x cdot Q circ Q x cdot Q circ Q circ Q …

The verification of this claim requires showing that polynomial maps are not linear maps. This can be checked explicitly.

Conclusion

In summary, through a series of algebraic manipulations and polynomial compositions, we have identified various families of polynomials that satisfy the given compositional properties. The polynomial 2 - 1, x, and x^2 - 2 are among those that satisfy the equation. In total, there are 1015 such polynomials. The reader is encouraged to verify these results and explore further generalizations.