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We have seen that the invariant ring of a G a -action on an affine variety need not be finitely generated as a k-algebra. But in many important cases, most notably in the linear case, the invariant ring is known to be finitely generated, and in these cases it is desirable to have effective means of calculating invariants. In this chapter, we consider constructive invariant theory for G a -actions, beginning with the classical linear case.
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