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    $\begingroup$ In principle one could construct a subring of a polynomial ring as the quotient ring of an evaluation homomorphism on a "bigger" polynomial ring. Whether this is computationally attractive might depend on just what you want to do with the subring once you have it. $\endgroup$ Commented Dec 2, 2015 at 16:07
  • $\begingroup$ @hardmath - I'll try that. (I.e. construct $f:R[T_1,\dots,T_m]\rightarrow A$ by mapping the $T$'s to my desired algebra generators, and then take Image($f$).) Do you know if Magma can intersect ideals of $A$ with Image($f$)? $\endgroup$ Commented Dec 3, 2015 at 0:44
  • $\begingroup$ I have some other ideas, but I thought you should propose a particular ring as the base for polynomials so we can compare approaches. Eg. polynomials over the integers, a field, or something more exotic. $\endgroup$ Commented Dec 3, 2015 at 1:07
  • $\begingroup$ @benblumsmith: Could you move your aside question to a question on SciComp meta? $\endgroup$ Commented Dec 3, 2015 at 6:23
  • $\begingroup$ @GeoffOxberry - done! meta.scicomp.stackexchange.com/questions/457/… $\endgroup$ Commented Dec 3, 2015 at 15:23