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This chapter deals with methods for computing the invariants of an arbitrary polynomial representation of the general linear group
). The main algorithm, to be presented in Sect. 4.6, is derived from Hilbert (1893). We will discuss Hilbert’s algorithm from the point of view of Gröbner bases theory. This chapter is less elementary than the previous three. While most of the presentation is self-contained, familiarity with basic notions of commutative algebra and representation theory will be assumed.