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Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$. Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\ldots f_m$.

Question: Is there an algorithm which give a generating set for the intersection $B\cap I$? In other words, if we consider the presentation of the commutative algebra $A/I$ arising from $I$, can we also get a nice presentation for its subalgebra $B/(B\cap I)$ ?

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    $\begingroup$ This is usually called "elimination theory". Consider first a bigger polynomial ring $K[X_1,\dots,X_n,Y_1,\dots,Y_m]$. Consider the bigger ideal $J=\langle g_1(X_1,\dots,X_n),\dots,g_r(X_1,\dots,X_n),Y_1-f_1(X_1,\dots,X_n),\dots,Y_m-f_m(X_1,\dots,X_n)\rangle$. Elimination theory gives generators for the ideal $J\cap K[Y_1,\dots,Y_m]$. The quotient of $K[Y_1,\dots,Y_m]$ by this ideal equals $B/(B\cap I)$. Macaulay2 computes this, among other computer algebra packages. $\endgroup$
    – Jason Starr
    Commented Mar 6, 2019 at 12:21
  • $\begingroup$ Many thanks for the answer. Can you give some more information about the algorithm used? How hard it is to do it by hand, for example? $\endgroup$
    – Ehud Meir
    Commented Mar 6, 2019 at 12:22
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    $\begingroup$ This is not an algorithm that is easy to follow by hand. An "elimination order" on the big polynomial ring is one where every monomial that has at least one positive exponent among $X_1,\dots,X_n$ is bigger than every monomial in $Y_1,\dots,Y_m$. For such a monomial order, for the associated reduced Groebner basis for $J$, those basis elements that lie entirely in $K[Y_1,\dots,Y_m]$ form a Groebner basis for the ideal $J\cap K[Y_1,\dots,Y_m]$. The algorithm to compute a Groebner basis is "Buchberger's algorithm". I recommend Cox-Little-O'Shea (I teach from that book). $\endgroup$
    – Jason Starr
    Commented Mar 6, 2019 at 12:32
  • $\begingroup$ Is there some reference for the isomorphism between $B / (B \cap I)$ and the quotient $K[Y_1, \ldots, Y_m]/(K[Y_1, \ldots, Y_m] \cap J)$? I would like to read further details. $\endgroup$
    – V.S.
    Commented Nov 16, 2021 at 17:21

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