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I'm having issues with a question in Artin, more specifically 11.3.3.e.

The question asks:

Find generators for the kernel of the following map: $\mathbb{C}[x,y,z] \to \mathbb{C}[t]$ given by $x \mapsto t$, $y \mapsto t^2$, $z \mapsto t^3$

Clearly we have $z-x^3$, $y-x^2$, and $z^2-y^3$ as elements of the kernel.

I'm not sure how to proceed. The book uses the division algorithm in the relevant examples, but it only does that for the two-variable case. I can't see how to apply it here and I also have a hard time understanding why it works. The other four parts were straightforward maps substituting variables for numbers, this is the first that deals with parametrization.

user26857
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Lost
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2 Answers2

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Hint. The kernel is generated by $z-x^3$, $y-x^2$. (In order to prove this use the division algorithm from an answer given here.)

Community
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I'll prove if $P(t,t^2,t^3)=0$ then $P(x,y,z)$ can be written $(z-x^3)f_1(x,y,z)+(y-x^2)f_2(x,y,z)$

Now let $P(x,y,z)=\sum a_{i,j,k} x^iy^jz^k$. Clearly after dividing by $z-x^3$ we'll have remainder something like $\sum a_{i,j}x^iy^j$ and again after diving by $y-x^2$ we'll have remainder of form $p_2(y)+xp_1(y)$. Now as per condition $p_2(t^2)+tp_1(t^2)=0$ but that means $p_1=p_2=0$.

So we get the desired result.

dragoboy
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