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+ h, if and only if there exist a sequence of indices ib i2, ...

A· U" ... ,a' u m ) < (fJ· U" ... ,fJ· u m ), where 0: . Ui is the lisual dot product in Qn. a. Prove that O. e. Let Ul, ... ,Um be vectors satisfying: for aU i, the first Uj sueh that Uji cl 0 satisfies Uji > O.

Let J be an ideal of k[Xl,'" ,xn ]. Assume that we are given two term orderings, say <, and <2 . Let {g" ... , g,} be a Grübner basis for J with respect to <, . Assume that lt<, (gi) = lt<, (gi), for i = 1, ... , t. Prove that {g" ... , g,} is also a Grübner basis for J with respect to <2 . 18. " ,g,} and G' = {g~ 1 • ' . ,g~/} be subsets of l of non-zero polynomials where we assume that G is a Grübner basis for J. Assume that for all gi E G we have gi = L}'~l aijgj where lp(gi) = max,:Si:s,,(lp(aiJ) lp(gj)).

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An Introduction to Grobner Bases (Graduate Studies in Mathematics, Volume 3) by Philippe Loustaunau, William W. Adams


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