Sequences Determining Mixed Multiplicities and Reductions of Ideals

On some multiplicity and mixed multiplicity formulas

“…, x p is a weak-(FC)-sequence of N with respect to (J, I). Then by [21,22](see [11,Proposition 3.3(iii) Since p is maximal, e(J [q−i] , I [i] ; N) = 0 if and only if 0 i p by [21]. Consequently by [21](see [11, by Remark 2.7.…”

“…[7,8,9,11,14,16,19,21,22,23,25]). In past years, using different sequences, one expressed mixed multiplicities into Hilbert-Samuel multiplicity, for instance: Risler-Teissier in 1973 [17] by superficial sequences and Rees in 1984 [13] by joint reductions; Viet in 2000 [21] by (FC)-sequences (see e.g.…”

A note on joint reductions and mixed multiplicities

“…In past years, the positivity and the relationship between mixed multiplicities and Hilbert-Samuel multiplicity of ideals have attracted much attention (see e.g. [2,3,4,7,8,9,13,14,17,19,20,21,22,23]). …”

The Euler-Poincaré Characteristic and Mixed Multiplicities

“…Using different sequences, one can express mixed multiplicities of ideals in terms of the Hilbert-Samuel multiplicity. For instance, in the case of n-primary ideals, Risler-Teissier [19] in 1973 showed that each mixed multiplicity is the multiplicity of an ideal generated by a superficial sequence and Rees [14] in 1984 proved that mixed multiplicities are multiplicities of ideals generated by joint reductions; for the case of arbitrary ideals, Viet [23] Note that [23] defines weak-(FC)-sequences in the condition I √ Ann R N (see, for example, [5,6,13,24,25,26,28,30,31]). In Definition 4.12, we omitted this condition.…”

“…Moreover, by using filterregular sequences, Manh and Viet [29] in 2013 characterized mixed multiplicities of multigraded modules in terms of the length of modules. In past years, the theory of mixed multiplicities has attracted much attention and has been continually developed (see, for example, [4,5,[11][12][13][14][15][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]). …”