Toward Mixed Multiplicities and Joint Reductions

Abstract

In the direction towards the question when mixed multiplicities are equal to the Hilbert–Samuel multiplicity of joint reductions, this paper not only generalizes Viet et al. (Proc Am Math Soc 142:1861–1873, 2014, Theorem 3.1) that covers the Rees’s theorem Rees (J Lond Math Soc 29:397–414, 1984, Theorem 2.4), but also removes the hypothesis that joint reductions are systems of parameters in Viet et al. (2014, Theorem 3.1). The results of the paper seem to make the problem of expressing mixed multiplicities into the Hilbert–Samuel multiplicity of joint reductions become closer.