Geodesic
Reciprocity laws and the stable rank of polynomial rings
Izvestiya. Mathematics , Tome 15 (1980) no. 3, pp. 589-623

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Reciprocity laws and Mennicke symbols for regular rings of arbitrary dimension are introduced and studied. The universal Mennicke symbol for the polynomial ring $k[X_1,\dots,X_n]$ relative to the principal ideal $(X_1^2-X_1)\cdots(X_n^2-X_n)$ is computed. The results obtained in this paper are applied to the problem of determining the precise value of the stable rank of polynomial rings and affine algebras over fields. Bibliography: 19 titles. 
@article{IM2_1980_15_3_a6,
     author = {A. A. Suslin},
     title = {Reciprocity laws and the stable rank of polynomial rings},
     journal = {Izvestiya. Mathematics },
     pages = {589--623},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {1980},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1980_15_3_a6/}
}
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A. A. Suslin. Reciprocity laws and the stable rank of polynomial rings. Izvestiya. Mathematics , Tome 15 (1980) no. 3, pp. 589-623. http://geodesic.mathdoc.fr/item/IM2_1980_15_3_a6/