Geodesic
Involutive division technique: some generalizations and optimizations
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part IV, Tome 258 (1999), pp. 185-207

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In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Gröbner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. We study dependence of involutive algorithms on the completion ordering. Based on properties of particular involutive divisions two computational optimizations are suggested. One of them consists in a special choice of the completion ordering. Another optimization is related to recomputing multiplicative and nonmultiplicative variables in the course of the algorithm. 
@article{ZNSL_1999_258_a9,
     author = {V. P. Gerdt},
     title = {Involutive division technique: some generalizations and optimizations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {185--207},
     publisher = {mathdoc},
     volume = {258},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_258_a9/}
}
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V. P. Gerdt. Involutive division technique: some generalizations and optimizations. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part IV, Tome 258 (1999), pp. 185-207. http://geodesic.mathdoc.fr/item/ZNSL_1999_258_a9/