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weighted initial ideal versus lex or graded reverse lex initial ideal

$$ $$ By imposing certain weights $\mathbf{w}$ on the variables, say, of a polynomial ring $k[x_1,\ldots, x_n]$, I read that we may obtain the initial ideal $In_{\mathbf{w}}(I)$ of an ideal $I$ with ...
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Isomorphism of ideals

I have a question related to this post: http://mathoverflow.net/questions/60412/generic-liftings-of-a-regular-sequence-on-the-initial-ideal Suppose $I$ and $J$ are ideals in $R=k[x_1,\ldots,x_n]$ ...