Let $I =\langle f\_1,\cdots,f\_m\rangle \subset K[x\_1,\cdots,x\_n]$be an ideal,
where $f\_k\in K[x\_1,\cdots,x\_n].$


$K[e\_1,\cdots,e\_n]$ the polynomial algebra generated by the elementary symmetric polynomials
 $e\_1,\cdots,e\_n\in K[x\_1,\cdots,x\_n].$


Is there any method(algorithm) to compute the K-algebra $I \cap K[e\_1,\cdots,e\_n]$ of the intersection of $I$ and $K[e\_1,\cdots,e\_n]? $


Since $K[e\_1,\cdots,e\_n]$ is not an ideal of $ K[x\_1,\cdots,x\_n],$ it fails to compute elimination ideal. Anyway, $I \cap K[e\_1,\cdots,e\_n]$ is an ideal of the ring 
$K[e\_1,\cdots,e\_n].$ What I want to do is to give the generating sets of this ideal by
polynomials in $K[e\_1,\cdots,e\_n].$



Finding generators of subalgebra of polynomial algebra $K[x_1,\cdots,x_n]$ that are invariant under the action of symmetric group
UPDATE: Sorry! As pointed out in a comment, my previous answer was incorrect. So I've edited my answer. The following simpler algorithm seems to me that it should work (at least assuming I'm understanding the question\dots).


From the geometric perspective, the inclusion map $k[e\_1, \dots, e\_n]\subseteq k[x\_1, \dots, x\_n]$ corresponds to the quotient $\phi: \mathbb A^n\to \mathbb A^n/S\_n$. The ideal $J:= I\cap k[e\_1, \dots, e\_n]$ is the contraction of the ideal $I$. Hence, geometrically, it seems to me that computing $J$ is equivalent to computing the ~~Zariski~~ scheme-theoretic closure of $\text{Spec}(k[x\_1, \dots, x\_n]/I)$ under the map $\phi$. 


So the ~~Zariski closure~~ Kernel of a Ring Map algorithm on page 84 of Greuel and Pfister's "A Singular Introduction to Commutative Algebra" would seem to be applicable. To apply the algorithm, you define a ring $R:=k[x\_1,\dots,x\_n, t\_1, \dots, t\_n]$ and then define an ideal $N\subseteq R$ by


$
N:=I+\langle t\_1-e\_1(\mathbf{x}), \dots, t\_n-e\_n(\mathbf{x}) \rangle
$


where $e\_i(\mathbf{x})$ is the $i$'th symmetric polynomial. Compute the elimination ideal $N\cap k[t\_1, \dots, t\_n]$, and say that it equals $\langle p\_1(\mathbf{t}), \dots, p\_r(\mathbf{t})\rangle$. Then $I\cap k[e\_1, \dots, e\_n]$ will equal the ideal $\langle p\_1(\mathbf{e}), \dots, p\_r(\mathbf{e})\rangle$.


I hope this helps.


