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Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and $C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated $\mathbf{Q}$-subalgebras of $A$ with explicit generators.

Q1: Is there a finite time (efficient) algorithm that allows one to say when is $B\simeq C$ as $\mathbf{Q}$-algebra?

Q2: Is there a finite time (efficient) algorithm that allows one to say when is $Frac(B)\simeq Frac(C)$?

Here $Frac(B)$ denotes the fraction field. In both questions I really mean isomorphic and not equal.

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  • $\begingroup$ I don't know much about algorithmic questions like this. But: it would probably help to specify whether you are given a set of generators for $B$ and $C$, or whether you merely have some implicit description of these subalgebras. $\endgroup$
    – MTS
    Commented Jun 9, 2012 at 0:06
  • $\begingroup$ Since $B$ and $C$ are both subalgebras of the same ring $A$, do you want to know whether they are isomorphic, or whether they are equal? $\endgroup$
    – David E Speyer
    Commented Jun 9, 2012 at 0:52
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    $\begingroup$ @David, here I really mean isomorphic. Equal would be "easy" since one may compute the relation ideal for $B$ and $C$ and then test for equality. $\endgroup$
    – Hugo Chapdelaine
    Commented Jun 9, 2012 at 2:42
  • $\begingroup$ @MTS, yes in both cases I have explicit sets of generators. $\endgroup$
    – Hugo Chapdelaine
    Commented Jun 9, 2012 at 2:46
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    $\begingroup$ mathoverflow.net/questions/21883/… $\endgroup$
    – M P
    Commented Jun 9, 2012 at 3:18

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