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I was wondering whether there is standard notation for a polynomial of a certain degree, say, $k$. That is, I want to be able to write $<standard \text{ } notation> $ instead of "..., where $p(x)$ is a polynomial of degree $k$."

Does such a thing exist? For example, would $p^k(x)$ do the trick? I don't really see it as self-evident that it would, but there might be better notation out there.

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    $\begingroup$ I think $p(x)\in \mathbb F_k[x]$ is standard. $\endgroup$ – Git Gud Jun 22 '14 at 23:11
  • $\begingroup$ @GitGud, I like your suggestion. $\endgroup$ – Ryker Jun 22 '14 at 23:26
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    $\begingroup$ @GitGud, that's possibly confused for a finite field if your $k$ is a prime power. $\endgroup$ – Kaj Hansen Jun 22 '14 at 23:28
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    $\begingroup$ @KajHansen You're right, I had never though about that. Still I think it's standard. Edit: Maybe it's standard only for fields of characteristic $0$ or maybe I got confused and the standard notation is $P_k\left(\mathbb F[x]\right)$. $\endgroup$ – Git Gud Jun 22 '14 at 23:29
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    $\begingroup$ @GitGud: I have never seen that notation and it seems quite confusing. For example, if your field is $\mathbb{Q}$ then does $\in \mathbb{Q}_2[x]$ mean a quadratic polynomial or does it mean a polynomial over the $2$-adic numbers? $\endgroup$ – Qiaochu Yuan Jun 23 '14 at 0:02
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I would write

$p\in\Bbb F[x],\ \deg(p)=k\,$.

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  • $\begingroup$ This is what I've always done. $\endgroup$ – Kaj Hansen Jun 22 '14 at 23:29
  • $\begingroup$ This seems non-standard. (Maybe it's standard in certain fields?) It seems far better to say "$p$, a polynomial, deg($p$) = $k$" or to use the summation notation. $\endgroup$ – Evgeni Sergeev Aug 26 '17 at 8:29
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How about p(x) = $\sum_{i=0}^k x^i a_i$? Maybe not what you're looking for, but the shorter than what you have already :)

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Polynomials (of degree $n$) are almost always written as $$a_0+a_1x+\cdots+a_nx^n$$ or more compactly as $$\sum_{k=0}^n a_kx^k.$$

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    $\begingroup$ I should've just gone with this, I guess, but then I feel compelled to add "for some $a_k \in \mathbb{R}$" or something of that nature :) Which is fine, of course, but I was hoping to get something that doesn't require any "where"'s. $\endgroup$ – Ryker Jun 22 '14 at 23:23

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