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First, define the polynomial ring.
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Consider the following almost general linear equations. They are not completely general, since their constant term, namely , is the same.
After projecting the variety defined by and into the parameter space given by the last 5 variables, you can see when such general linear equations have solutions after specializing the last 5 variables.
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There are 9 regular systems defining the image cs of the projection. To remove common parts of these regular systems, use MakePairwiseDisjoint.
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Now, there are 10 components.
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Notice that some components have split during the redundancy removal.