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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 $\cap_{i=1}^n R_i$?

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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).

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