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Its easy to see why a set of $n$ polynomials $\{p_1\cdots p_n\}$ may have fewer than $n$ Groebner bases. For example, if $p_n=p_1*p_2+p_3$ then $\{p_1\cdots p_{n-1}\}$ and $\{p_1\cdots p_n\}$ generate the same ideal. As a result I would expect it to have $n-1$ or fewer groebner bases.

However using maple, I occasionally see systems where $n$ polynomials have more than $n$ GB. Could anyone explain why? and give a simple example?

Abijah
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  • Well, not sure what you mean. Do you refer to a fixed term order? Then there is a uniquely determined reduced GB. For each GB, adding a polynomial of the ideal in question gives another GB. – Wuestenfux Oct 04 '19 at 12:59
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    @Wuestenfux: I think the O.P. means ‘why a Groebner basis may have more elements that the number of given generators of the ideal’. – Bernard Oct 04 '19 at 14:02

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