# Quick question on polynomial notation (from the perspective of rings)

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:

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~