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I am reading some chapters about polynomials in the context of ring theory and am trying to develop some familiarity with the concept.

After reading this post, What actually is a polynomial?, I certainly have a better understanding. However, one of the answers had an example where I stumbled on the notation justification...here is an excerpt from said answer:


Notation


My confusion is from the very last line...where the author states that:

$$ (0, -2y^2 + 6y^3 , \color{#c00}{4y^3}) = ((0), (0, 0, -2, 6), \color{#c00}{(0, 4)}) $$

The left side of the equation makes sense, for if $X$ is the symbol of interest, then the $x^0$ coefficient has a $0$ in front of it, the $x^1$ symbols have $-2y^2 +6y^3$ coefficients in front of them, etc.

The right side of the equality is where I am stuck. I assume that this is now rewriting the left side notation by embedding $Y$ as the symbol of interest.

For example, in examining $-2y^2+6y^3$, the leading coefficient of $y^0$ is $0$, of $y^1$ is $0$, of $y^2$ is $-2$ and of $y^3$ is $6$...which would produce the notation of $(0,0,-2,6)$.

Following the same notation strategy, shouldn't $\color{#c00}{4y^3}$ be expressed as $(0,0,0,4)$ instead of $\color{#c00}{(0,4)}$? Thanks~

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You are correct - and have therefore understood the abstraction involved.

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