TITLE: What is the name for this parameter of polynomial?
QUESTION [0 upvotes]: Could you please help me?
In the polynomial $p(x) = a_k x^k + a_{k+1} x^{k+1} + ... + a_n x^n$ ($a_k, a_n \neq 0$), we say that $n$ is a "degree" of this polynomial, deg $p = n$.
Is there any special word or notation for $k$?
I did not find the answer with the help of the Google.

REPLY [2 votes]: As far as I know, there is no special name for $k$.
$k$ is the order$^1$ of the zero at $x=0$ (because $a_k\neq0$).
You could say that $k$ is the minimum degree of all monimials comprising $p$, but that is quite long-winded.  In particular if you are nor referring to monomials anywhere else.
Also $k$ is the index of the lowest or first (by degree) non-zero coefficient of $p$.
In the context of valued fields it can be used to define a non-trivial valuation of $p$, but outside that context this term might be quite confusing or has not been defined.
$^1$ Where order = 0 means that 0 is not a zero of $p$.
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