finite codimensional subalgebra of $\mathbb C[z]$

Asked Apr 21 '21 at 05:30

Active Apr 21 '21 at 05:47

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If $p_1, p_2$ are two polynomials in $\mathbb C[z]$, then when $\mathbb C[p_1, p_2]$ is finite codimensional in $\mathbb C[z]$? Are there some sufficient conditions (or NASC) on $p_1, p_2$?

It is apparently clear that if $p_1 = p_2$, $\deg p_1>1$, then it is not finite codimensional. For example, take $\mathbb C[z^2]$.

edited Apr 21 '21 at 05:47

Jyrki Lahtonen

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asked Apr 21 '21 at 05:30

belsam

Certainly not always, e.g. $p_1=p_2=z^2$. – Joshua P. Swanson Apr 21 '21 at 05:36
I recall seeing a related discussion either here or on MathOverflow. The only thing I could find right away was particular to codimension one. – Jyrki Lahtonen Apr 21 '21 at 05:37
@Joshua P. Swanson, I agree. I edited my question before I see your comment. – belsam Apr 21 '21 at 05:44
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IIRC there is either a result or a conjecture that the counterexamples have the form $p_1=r_1(q)$, $p_2=r_2(q)$ for some polynomials $r_1,r_2,q$, $\deg q>1$. These trivially have infinite codimension, but the other direction is anything but obvious (to me at least). – Jyrki Lahtonen Apr 21 '21 at 05:45
@JyrkiLahtonen, would it be possible to suggest some reference around this question. – belsam Apr 21 '21 at 05:55
Not from me, I'm afraid. That's why the recollections are vague. – Jyrki Lahtonen Apr 21 '21 at 05:56

finite codimensional subalgebra of $\mathbb C[z]$

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