Rings, ideals and Groebner basis

[AlphaNumeric]

How does a Goebner basis relate to Ideals and how does it help solve otherwise extremely complex systems of equations?

Part of what I'm working on involves trying to solve V=0=\partial_{i} V 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 0=\partial_{i} V 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 0=\partial_{i} V? Are the roots of the polynomials of a Groebner basis exactly the 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!

 

[Hurkyl]

If you supply a notion of complexity for polynomials, and a list of polynomials that generates an ideal I, then a reduced Gröbner basis is the least complex list of polynomials that generates I.


If V is the solution set to your polynomials f_i, then the ideal

I = (f₁, ..., f_n)​

is a set of polynomials whose solution set is exactly V. If I is radical, then I is actually the entire collection of polynolmials whose zero set includes V.

The upshot is that a Gröbner basis algorithm will find the least complex list of polynomials that generate I, and thus a simplified set of polynomials that define exactly V. One drawback is that "least complex" is not "simple" -- if V is complicated, or you make a poor choice for the notion of complexity, you will not get a "simple" Gröbner basis.


But, there are some things you can do about it. Magma, at least, can compute a "prime decomposition" -- Gröbner bases tend to be very complicated when V has many algebraic components. A prime decomposition will split V into its individual components, which will have better Gröbner bases.

And I know one of the term orders (the thing that defines "complexity") will guarantee that the last polynomial in the Gröbner basis will only involve a particular variable of your choice (if such a thing can even be done) -- so you can manually split the problem into cases by considering each root of this polynomial as a separate problem.