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{\bf Gr\"obner Bases with respect to Several Orderings\\
and Multivariable Hilbert Polynomials}

Alexander Levin 

Department of Mathematics, The Catholic University
of America

Washington, D. C. 20064


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{\bf Abstract}

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Let $R = k[X]$ be a ring of generalized polynomials in a set of 
indeterminates $X$ over a field $k$, and let a partition of $X$ into 
$p$ disjoint subsets be fixed. Furthermore, let $E$ be a finitely 
generated free $R$-module. We consider $p$ orderings of the set of 
terms of $E$ that naturally correspond to the partition of $X$ and 
develop a Gr\"obner basis technique that involves these orderings. 
As an application we give a method of computation of 
Hilbert polynomials in several variables.

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