Additive polynomial

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In mathematics, the additive polynomials are an important topic in classical algebraic number theory.

Let ${\displaystyle k}$ be a field of prime characteristic ${\displaystyle k}$. A polynomial ${\displaystyle P(x)}$ with coefficients in ${\displaystyle k}$ is called an additive polynomial, or a Frobenius polynomial, if

$${\displaystyle P(a+b)=P(a)+P(b)}$$

as polynomials in ${\displaystyle a}$ and ${\displaystyle b}$. It is equivalent to assume that this equality holds for all ${\displaystyle a}$ and ${\displaystyle b}$ in some infinite field containing ${\displaystyle k}$, such as its algebraic closure.

Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition that ${\displaystyle P(a+b)=P(a)+P(b)}$ for all ${\displaystyle a}$ and ${\displaystyle b}$ in the field.^[1] For infinite fields the conditions are equivalent,^[2] but for finite fields they are not, and the weaker condition is the "wrong" as it does not behave well. For example, over a field of order ${\displaystyle q}$ any multiple ${\displaystyle P}$ of ${\displaystyle x^q-x}$ will satisfy ${\displaystyle P(a+b)=P(a)+P(b)}$ for all ${\displaystyle a}$ and ${\displaystyle b}$ in the field, but will usually not be (absolutely) additive.

The polynomial ${\displaystyle x^p}$ is additive.^[1] Indeed, for any ${\displaystyle a}$ and ${\displaystyle b}$ in the algebraic closure of ${\displaystyle k}$ one has by the binomial theorem

$${\displaystyle (a+b{)}^p={\displaystyle \sum _{n=0}^p}(\genfrac{}{}{0ex}{}{p}{n})a^n{b}^{p-n}.}$$

Since ${\displaystyle p}$ is prime, for all ${\displaystyle n=1,\dots ,p-1}$ the binomial coefficient ${\displaystyle (\genfrac{}{}{0ex}{}{p}{n})}$ is divisible by ${\displaystyle p}$, which implies that

$${\displaystyle (a+b{)}^p\equiv a^p+b^{p}modp}$$

as polynomials in ${\displaystyle a}$ and ${\displaystyle b}$.^[1]

Similarly all the polynomials of the form

$${\displaystyle {\tau }_p^n(x)=x^{p^n}}$$

are additive, where ${\displaystyle n}$ is a non-negative integer.^[1]

It is quite easy to prove that any linear combination of polynomials ${\displaystyle {\tau }_p^n(x)}$ with coefficients in ${\displaystyle k}$ is also an additive polynomial.^[1] An interesting question is whether there are other additive polynomials except these linear combinations. The answer is that these are the only ones.^[3]

One can check that if ${\displaystyle P(x)}$ and ${\displaystyle M(x)}$ are additive polynomials, then so are ${\displaystyle P(x)+M(x)}$ and ${\displaystyle P(M(x))}$. These imply that the additive polynomials form a ring under polynomial addition and composition. This ring is denoted^[4]

$${\displaystyle k\{{\tau }_p\}.}$$

This ring is not commutative unless ${\displaystyle k}$ is the field ${\displaystyle {\mathbb{𝔽}}_p=\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}}$ (see modular arithmetic).^[1] Indeed, consider the additive polynomials ${\displaystyle ax}$ and ${\displaystyle x^p}$ for a coefficient ${\displaystyle a}$ in ${\displaystyle k}$. For them to commute under composition, we must have

$${\displaystyle (ax{)}^p=a{x}^p,}$$

and hence ${\displaystyle a^p-a=0}$. This is false for ${\displaystyle a}$ not a root of this equation, that is, for ${\displaystyle a}$ outside ${\displaystyle {\mathbb{𝔽}}_p.}$^[1]

Let ${\displaystyle P(x)}$ be a polynomial with coefficients in ${\displaystyle k}$, and ${\displaystyle \{w_1,\dots ,w_m\}\subset k}$ be the set of its roots. Assuming that the roots of ${\displaystyle P(x)}$ are distinct (that is, ${\displaystyle P(x)}$ is separable), then ${\displaystyle P(x)}$ is additive if and only if the set ${\displaystyle \{w_1,\dots ,w_m\}}$ forms a group with the field addition.^[5]

• Drinfeld module
• Additive map

• Weisstein, Eric W.. "Additive Polynomial". http://mathworld.wolfram.com/AdditivePolynomial.html. 

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