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Consider a polynomial ring with two variables
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Consider two regular chains in R
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These two regular chains share a common solution. The union of their zero sets can be made disjoint. In other words we can replace these two regular chains by another set of regular chains such that the two sets describe the same solutions and the second one consists of pairwise disjoint zero sets of regular chains. This is done as follows
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| (4) |
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| (5) |