Suppose we are given two polynomial rings $R\_1$ and $R\_2$ by presenting their generators, $S\_1$ and $S\_2$, where $S\_i$ are finite set of $m\_i$ variables, $i.e.$, $S\_i\subset P[x\_1,x\_2,\cdots,x\_{m\_i}]$. $R\_i$ are generated from $S\_i$ by the usual addition, subtraction, and multiplication. 


The goal is to compute the intersection ring of $R\_1$ and $R\_2$ by given some generating set of $R\_1\cap R\_2$. Is $R\_1\cap R\_2$ still finitely generated?


How about general case that there are $n$ rings, and $S\_i$ only consists of homogeneous polynomials with degree $l\_i$, can we bound the degree of some generating set of $\cap\_{i=1}^n R\_i$?



Bound the degree of the generator of polynomial ring
Without homogeneity assumptions, the intersection need not be finitely generated. This is discussed in <http://arxiv.org/abs/1301.2730>. (See also <http://www.emis.de/journals/BAG/vol.43/no.2/b43h2bay.pdf> which is referred to in that arXiv preprint).


