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For example, are forms like $\sin(\arcsin(x))$ considered polynomials?

Yes it simplifies to $x$, but $x$ and $\sin(\arcsin(x))$ have very different domain and ranges.

Math Attack
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N Kabir
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    Depends what you mean by a polynomial here. Your example is a function which on its domain coincides with a polynomial function. – Mark Apr 15 '23 at 17:09
  • I'd say "no". I'd describe that as "a function that reduces/simplifies to a polynomial (on its range, which isn't all of $\mathbb R$)". – JonathanZ Apr 15 '23 at 17:13
  • @Mark Isn't polynomial a well defined term in math? Or, is it more logical to consider that as a polynomial or not? – N Kabir Apr 15 '23 at 17:16
  • An expression is polynomial (in $x$) if it is of the form $a_{0}+a_{1}x+\dots a_{n}x^n$ where the coefficients do not depend on $x$. The function generated by a given expression is polynomial if there exists a polynomial expression that gives us the same function. So $sin(arcsin(x))$ is not a polynomial, but it does generate a polynomial function. – user3257842 Apr 15 '23 at 17:23
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    For $~x \in [-1,1],~$ let $f(x) = x.~$ Is $~f(x)~$ considered a polynomial? – user2661923 Apr 15 '23 at 17:28
  • Good question... – tryst with freedom Apr 18 '23 at 22:17
  • "polynomial" is an overloaded term - having varying (closely related) denotations depending on the context (similar overloaded terms are "number", "vector", "power series", "fraction", etc). Surely most all of these denotations are discussed in prior answers. Did you search fpr answers before posing your question? – Bill Dubuque Apr 18 '23 at 23:07
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    Related – tryst with freedom Apr 19 '23 at 00:52
  • is the expression $A^2+A+3\cdot Id_{n\times n}$ a polynomial for a matrix $A\in Mat_{n\times n}$? yes, it is. What do we actually mean, if we talk about a polynomial? It depends on the context. But in general, I would prefer, to use the term "polynomial", if a mathematical object, can be represented or described as an (finite) arithmetic expression consisting of operations addition and multiplication of variables related to this object. I mean, you will find this uncertainties in various fields of mathematics. Sometimes Mathematicans talk about the equation $f(x)=0$ where $f$ vanishes. – cogitoergoboom Apr 19 '23 at 11:50
  • I think that maybe your question is more related which functions that could be described through a power series, like functions which are equivalent to their Taylor expansion when infinite terms are considered. Actually this is just a small number of function within the whole possibility, since it require to them to be infinitely times differentiable, which is not the case, for example, of flat functions. Maybe rephrase your question to be more specific in what you need to understand now you got many comments. – Joako Apr 19 '23 at 20:19
  • Related: https://math.stackexchange.com/questions/2185587/what-actually-is-a-polynomial/2185648#2185648 – Ethan Bolker Apr 23 '23 at 20:57

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A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division. It does not include trigonometric functions.

$\sin(\arcsin(x))$ is a trigonometric function and is not an expression of any power of $ x $, that is why it is not a polynomial.

Trigonometric functions are periodic (a function that repeats its values at regular periods) and cannot be described by a polynomial.

HeroZhang001
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Jonathan Xu
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