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This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$, namely, $f=u(h)$ and $g=v(h)$ for some $u(t),v(t) \in k[t]$; the answer is positive.

Is it possible to replace the field $k$ by an integral domain $D$? Namely: If $f,g \in D[x,y]$ are two algebraically dependent polynomials over an arbitrary integral domain $D$, is it true that there exists a polynomial $h \in D[x,y]$ such that $f,g \in D[h]$?

Denote the field of fractions of $D$ by $Q(D)$. It is clear that if $f,g \in D[x,y] \subset Q(D)[x,y]$ are two algebraically dependent polynomials over $D$, then from the above question there exists a polynomial $h \in Q(D)[x,y]$ such that $f,g \in Q(D)[h]$, namely, $f=u(h)$ and $g=v(h)$ for some $u(t),v(t) \in Q(D)[t]$.

I do not see why, for example, $D[x][y] \ni f=u(h)=u_mh^m+\cdots+u_1h+u_0$ should imply that $h \in D[x,y]$ and $u_j \in D$ (changing variables does not seem to help, namely if the leading term is $cy^l$, with $c \in Q(D)$).

Any comments are welcome.

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  • $\begingroup$ I guess that there exists also a counterexample with $D=k[t^2,t^3]$, the simplest non-normal integral domain. It seems that $f=(tx+ty)^2=t^2(x+y)^2 \in D[x,y]$ and $g=(tx+ty)^3=t^3(x+y)^3 \in D[x,y]$ is a counterexample. $\endgroup$ – user237522 Nov 1 '18 at 0:37
  • $\begingroup$ Another example in $k[t^2,t^3][x,y]$: $f=t^2x+t^3y$, $g=t^3x+t^4y$. We have, $\operatorname{Jac}(f,g)=\operatorname{Jac}(t^2x+t^3y,t^3x+t^4y)=t^2t^4-t^3t^3=0$. Notice that $g=tf$, so there is no $h \in k[t^2,t^3][x,y]$ such that $f,g \in k[t^2,t^3][h]$. $\endgroup$ – user237522 Nov 1 '18 at 17:08
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No. Choose a field $k$ and $D=k[u^2,u^3,v^2,v^3,uv]\subset k[u,v]$, so $D$ is a noetherian domain. In $D[x,y]$, choose $f=(ux+vy)^2$ and $g=(ux+vy)^3$; they are clearly algebraically dependent (but $ux+vy\notin D[x,y]$). Write $K=k(u,v)=\mathrm{Frac}(D)$.

Claim: there is no $P\in D[x,y]$ such that $f,g\in D[P]$.

By contradiction, let $P\in D[x,y]$ such that $f,g\in D[P]$. So $f=r_1(P)$, $g=r_2(P)$ with $r_1,r_2\in D[t]$. Since $f^3=g^2$ and $K[t]$ is a UFD, there exists $r\in K[t]$ such that $r_1=r^2$ and $r_2=r^3$. So $r(P)^2=f$. So $r(P)=\pm (ux+vy)$; up to change $(u,v)$ to $(-u,-v)$, let us suppose $r(P)=ux+vy$. This implies that $r\in K[t]$ and $P\in K[x,y]$ have degree 1. Write $r=at+b$ and $P=cx+dy+e$; then $r(P)=acx+ady+ae+b=ux+vy$. So $ac=u$, $ad=v$, $ae+b=0$.

So $c,d$ are nonzero; then $c/d=u/v$, hence $cv=du$. Since $c,d\in k[u,v]$, we can write $c=uq$ and $d=vq$ with $q\in k[u,v]$. We have $aq=1$. Since $r^2\in D[t]$, we have $a^2\in D\subset k[u,v]$. So $a^2$ is invertible in $k[u,v]$, and hence $a\in k^*$. So $u=a^{-1}c\in D$, a contradiction.

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  • $\begingroup$ Thank you very much! What if $D$ is an integral domain, for example $D=\mathbb{Z}$? $\endgroup$ – user237522 Dec 3 '17 at 13:58
  • $\begingroup$ @user237522 my $D$ is an integral domain... $\endgroup$ – YCor Dec 3 '17 at 14:03
  • $\begingroup$ Oh sorry I meant a UFD. $\endgroup$ – user237522 Dec 3 '17 at 14:04
  • $\begingroup$ Still my $D$ is indeed not a normal domain, which sounds like a natural requirement to look at. UFD is a stronger requirement, and PID even stronger. Non-normality sounds central in my argument... $\endgroup$ – YCor Dec 3 '17 at 14:05
  • $\begingroup$ I even do not mind to assume in my question that $D=\mathbb{Z}$, the simplest UFD I know; can you find a counterexample for $\mathbb{Z}$? $\endgroup$ – user237522 Dec 3 '17 at 14:08

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