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Definition: Let $0\leq i \leq n$ and $A_i,B_i \in \mathbb{E}$. Then we call the polynomial $S(X)=\sum_{i=0}^{n-1}B_i X^{{q}^i}+A$ the univariate representation of the affine transformation $S(X)$.

$\mathbb{E}$ is an extension of $\mathbb{F}$ and $|\mathbb{F}|=q$

My question is Why, in $S(X)$, the elements $X$ have exponent $q^i$, What's mean that?

juaninf
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1 Answers1

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The point is that $X\mapsto X^q$ is linear in $\Bbb E$ (over $\Bbb F$, at least), whereas the other polynomials are not linear, so they couldn't form an affine transformation.

Berci
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