Geodesic
On the computation of multiplicity by the reduction of dimension
Acta mathematica Universitatis Comenianae, Tome 78 (2009) no. 2 
E. Boďa; D. Jašková. On the computation of multiplicity by the reduction of dimension. Acta mathematica Universitatis Comenianae, Tome 78 (2009) no. 2. http://geodesic.mathdoc.fr/item/AMUC_2009_78_2_a4/
@article{AMUC_2009_78_2_a4,
     author = {E. Bo\v{d}a and D. Ja\v{s}kov\'a},
     title = {On the computation of multiplicity by the reduction of dimension},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2009},
     volume = {78},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2009_78_2_a4/}
}
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Voir la notice de l'article provenant de la source Comenius University

In this short note we describe one method for the computation of the Samuel multiplicity of the polynomial ideals and prove a formula for the multiplicity of the ideal α ixiai – β i+1 x i+1 bi+1 ; i =1,. . ., n ) . R in R (with the convention x n+1 = x 1 , β n+1 = β 1 , b n+1 = b 1 ), where ( R, m ) = k [ x 1 , x 1 , . . , x n ] (x1, x1, . . ,xn) is a local polynomial ring over an algebraic closed field k .