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Ideals, bifiltered modules and bivariate Hilbert polynomials. (English) Zbl 1126.13017

Author’s abstract: Let \(R\) be a ring of polynomials in \(m+n\) variables over a field \(K\) and \(I\) be an ideal in \(R\). Furthermore, let \((R_{rs})_{r,s\in Z}\) be the natural bifiltration of the ring \(R\) and let \((M_{rs})_{r,s\in Z}\) be the corresponding natural bifiltration of the \(R\)-module \(M=R/I\) associated with the given set of generators induced by Levin.
The author shows an algorithm for constructing a characteristic set \(G=\{g_1,\dots,g_s\}\) of \(I\) with respect to a special type of reduction introduced by Levin, that allows one to find the Hilbert polynomial in two variables of the bifiltered and bigraded \(R\)-module \(R/I\). This algorithm can be easily extended to the case of bifiltred \(R\)-submodules of free \(R\)-modules of finite rank \(p\) over \(R\).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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References:

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