How does a Goebner basis relate to Ideals and how does it help solve otherwise extremely complex systems of equations?(adsbygoogle = window.adsbygoogle || []).push({});

Part of what I'm working on involves trying to solve [tex]V=0=\partial_{i} V[/tex] for V a quotient of polynomials in several variables. This paper talks about using the Groebner basis to aid in such work but I'm a little fuzzy on the details. I see how Equation 17 is got, it's a decomposition of all possible ways that [tex]0=\partial_{i} V[/tex] is true in relation to the f's. But I don't see how the Groebner basis relates to helping. The paper says that it effectively decomposes a bunch of horrific equations into much more managable ones. For instance, Equations 33 generate V in (34) which then produces two ideals in (35). At the components in the two ideals of (35) the equations whose roots are the values which match the roots of [tex]0=\partial_{i} V[/tex]? Are the roots of the polynomials of a Groebner basisexactlythe same as the original Ideal, none added, none taken away (except for redundant ones dropped when taking the radical).

I'm a bit confused because the few systems I've tried with Mathematica and Singular seem to give a Groebner basis which is often just as complex as the original system and sometimes completely different, containing terms which weren't even in the original Ideal?!

If I've said anything which just doesn't make sense, I'm not suprised, I thought I'd left rings and ideals behind long ago then theoretical physics pulls another pure maths application out of the bag!

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# Rings, ideals and Groebner basis

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