Rings, ideals and Groebner basis

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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.

 

[tacman]

Rings, ideals, and Groebner basis are all concepts in abstract algebra, specifically in the field of commutative algebra. A ring is a mathematical structure that consists of a set of elements and two operations, usually addition and multiplication, that satisfy certain properties. In the context of polynomials, a ring is a set of polynomials with coefficients from a certain field, such as the field of real or complex numbers. Ideals are subsets of a ring that satisfy certain properties, and they can be thought of as generalizations of the concept of divisibility in integers. An ideal is a set of polynomials that can be multiplied by any element in the ring and still remain in the ideal.

A Groebner basis is a set of polynomials that generate an ideal in a specific way, called the Groebner basis algorithm. This algorithm takes a set of polynomials and produces a minimal set of polynomials that generate the same ideal. This means that any polynomial in the ideal can be written as a combination of the polynomials in the Groebner basis.

So, how does a Groebner basis relate to ideals? A Groebner basis is a special set of polynomials that generates the ideal in a specific way. It is a powerful tool because it allows us to simplify and solve complex systems of equations involving polynomials. This is because, as you mentioned, the Groebner basis effectively decomposes a set of equations into simpler ones.

In the context of solving V=0=\partial_{i} V for V a quotient of polynomials, a Groebner basis can help by reducing the problem to solving a set of simpler equations. This is because the Groebner basis algorithm produces a minimal set of polynomials that generate the same ideal, and this set is often much smaller and more manageable than the original set of equations.

As for your confusion about the roots of the polynomials in a Groebner basis, they are not necessarily the same as the roots of the original ideal. The Groebner basis algorithm does not guarantee that the roots will be the same, but it does guarantee that the ideal generated by the Groebner basis will be the same as the original ideal. This means that the solutions to the simpler equations produced by the Groebner basis will also be solutions to the original system of equations.

In summary, a Groebner basis is a powerful tool in commutative algebra that helps simplify