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I don't understand a step in this proof of the Hilbert Basis Theorem. Here is the proof Planeth Math.

I don't understand why $ \mathrm{deg} (f_{N+1}-g)< \mathrm{deg}(f_{N+1}) $. This can only happen if $ g $ has the same degree as $ f_{N+1} $ and also the same leading term, but I don't know why.

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    Did you follow the construction of $g$? More specifically, the elements $u_k$ and the exponents $v_k$ were not chosen randomly.2012-03-20
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    They constructed $g$ in such a way that it cancels the lead coefficient of $f_{N+1}$.2012-03-20
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    yes you are right, i saw it, thanks2012-03-20

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