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We first offer a fast method for calculating the Gelfand-Kirillov dimension of a finitely presented commutative algebra by investigating certain finite set. Then we establish a Groebner-Shirshov bases theory for bicommutative algebras, and show that every finitely generated bicommutative algebra has a finite Groebner-Shirshov basis. As an application, we show that the Gelfand-Kirillov dimension of a finitely generated bicommutative algebra is a nonnegative integer.

In this paper, we elaborate Gr\"obner-Shirshov bases method for Leibniz (super)algebras. We show that there is a unique reduced Gr\"obner-Shirshov basis for every (graded) ideal of a free Leibniz (super)algebra. As applications, we construct linear bases of free metabelian Leibniz superalgebras and new linear bases of free metabelian Lie algebras. We present a complete characterization of extensions of a Leibniz (super)algebra by another Leibniz (super)algebra, where the former is presented by generators and relations., Comment: 38 pages