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I'm confused at the exact nature of a polynomial in a set or ring, is a polynomial its own kind of object? We use indeterminates so we are interested in the structure, not the number, it could be an expression, however, say we have $X^2+3X+4$ and $4+3X+X^2$ or $X^2+(2+1)X+4$ these are all different expressions but the same polynomial, which suggests that polynomials are at a higher level than expressions, how do we understand this?

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    $\begingroup$ A polynomial with coefficients in a ring $R$ is a sequence $(a_0, a_1, a_2, \ldots)$ of elements of $R$ which has only finitely many nonzero entries. (Usually this polynomial would be written as $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$.) $\endgroup$
    – littleO
    Aug 8, 2022 at 8:38
  • $\begingroup$ I'm really not a fan of the sequential definition of polynomials, because it doesn't capture what a polynomial is supposed to be. This old answer of mine to a similar question might be of interest to you: math.stackexchange.com/questions/3948623/… $\endgroup$ Aug 8, 2022 at 10:52
  • $\begingroup$ Also the chosen base $1,x,x^2,\cdots,x^n$ is not mandatory, you can express polynomials in other interesting bases, the binomial one for instance $1,x,\frac 12x(x-1),\cdots,\binom{x}{n}$. What's important it's that it is described by $n$ coefficients. $\endgroup$
    – zwim
    Aug 8, 2022 at 12:20
  • $\begingroup$ Your examples of three different expressions for the same objects can be done with every mathematical object, like $0=1-1=\sin(0)=\ln(1)=...$. More interesting is that in a finite field of coefficients, different polynomials could induce the same polynomial function. Example: with $R=\mathbb Z/2Z$, $X^2+1$ and $X+1$ are different polynomials that induce the same polynomial function. $\endgroup$
    – Taladris
    Aug 8, 2022 at 12:21

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As pointed out in the comments polynomials are bascailly just sequences with it's elements in the Ring (at least that's how they were defined in my lectures). A common definition could be as follows:

A polynomial is a sequence $(a_i)_{i \in \mathbb{N}_0} \subseteq R$ (where R is a Ring or Field) with only finite many $a_i \neq 0$.

With this one can define the addition and multiplication as we know them. Addition would just be the addition of the sequence Elements and multiplication would be defined with a 'convolution' (not sure about the terminology, in German it's "Faltungsprodukt")

Now in order simplify working with polynomials one defines the notation of Polynomials as a sum of powers of x with the coefficients being the elements in the sequence.

This could be as follows: Let $p=(a_j)_{j \in \mathbb{N}_0}$ be a polynomial, then we define: $$ p := \sum_{j \in \mathbb{N}_0} a_j x^j $$ (Note that here it is important that only finite many $a_i \neq 0$ for the sum to exist)

So really the notation we know is just a notation and as such there are possibilities of redundancy as you have pointed out.

However it seems to have turned out that the above is quite a useful notation which is why it is used rather than the notation as sequences.

So if one was very pedantic with the definition above one would have to simplify your example terms and would only then be able to talk about those terms as polynomials and would see that they are indeed the same polynomial.

At last all this also just depends on the definition of polynomials that is used and this would just be my take on it with the definition I have been taught.

I hope this helped.

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