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Abstract: Commutative Grobner Bases are a well established technique with many applications, including polynomial solving and constructive approaches to commutative algebra and algebraic geometry. Noncommutative Grobner Bases are a focus of much recent research activity. For example, combining invariant theory and elimination theory, or elimination in moving frames of partial differential operators invariant under an equivalence group, requires the use of noncommutative Grobner bases. This paper presents theory and algorithms for noncommutative Grobner bases in Poincar\'e-Birkhoff-Witt extensions. These extension rings generalize the previous domains over which non-commutative Grobner bases have been applied. Our approach to noncommutative Grobner bases differs from previous work which assumes that the coefficients are from a field or commutative ring. In applications such as Cartan's method of moving frames, this is not the case, and the theory that we present can be applied.
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