On (De)homogenized Gröbner Bases

Abstract

Let $K$ be a field and $R={\oplus }_{p\in \mathbb{N}}{R}_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gröbner basis theory. It is proved that there is a one-to-one correspondence between the set of Gröbner bases in $R$ and the set of dh-closed homogeneous Gröbner bases in the polynomial algebra $R[t]$; and that the similar result holds true if $R$ and $R[t]$ are replaced respectively by the free algebra $K<X_1,...,X_n>$ and the free algebra $K<X_1,...,X_n,T>$. Moreover, it is shown that dh-closed graded ideals in $R[t]$ and $K<X_1,...,X_n,T>$ can be realized by dh-closed homogeneous Gröbner bases. The latter result indeed tells us that algebras defined by dh-homogeneous Gröbner bases can be studied as Rees algebras effectively via more simpler algebras as demonstrated in ([7], [8]).

This is a new version of arXiv:0907.0526v1 combined with arXiv:0907.2483v1. 23 pages

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