[ur-whale]

Naive question, but gotta ask: can Gröbner basis get over the "in general, a polynomial of degree >=5 can't be solved algebraically" hump?

Given that the above is a theorem (from Galois IIRC) I'd assume the answer is no.

And given that any system of polynomial equations in multiple variables - even of low degree - eventually subsume to a univariate polynomial of very high degree , then ... what are Gröbner actually useful for?

[jjtheblunt]

You know how someone walking down the street might give you a collection of vectors over a field, and ask questions about the subspace the vectors generate? You might make an orthonormal basis set of vectors which generate the same subspace, as a means to more easily answer questions about the original vectors.

Groebner bases play a role similar to a minimal orthonormal basis for a set of arbitrary vectors in a vector field, except the set of things to be linearly combined, and about which the linear combinations will be asked questions, are not tuples of floats (vectors) but instead are polynomials.

https://a.co/d/iuciOR5

is a famous book that's very nice (or was 25 years ago in earlier editions) discussing this topic and the various answers to your question.

[_dain_]

>You know how someone walking down the street might give you a collection of vectors over a field, and ask questions about the subspace the vectors generate?

they can be very insistent but legally you don't have to answer them.

[_a_a_a_]

> You know how someone walking down the street might give you a collection of vectors over a field, and ask questions about the subspace the vectors generate

what on earth...? I can't make sense of the last 2/3ds and with the first 1/3rd it is just weird. What does any of it mean anyway?

[NtochkaNzvanova]

Most of these concepts (except Groebner bases) are basic stuff you would learn in a first course in linear algebra. If you don't know what they mean, this post is likely not relevant to you.

[jjtheblunt]

A simple analogy would be that a groebner basis, for a set of given polynomials, is similar to an orthonormal basis for a set of vectors.

[jjtheblunt]

( Well doesn’t happen quite like that usually, unless the Someone is your math phd research Prof. looking to generalize an idea. )

[enedil]

Galois actually proved, that you cannot express the solutions as a set of certain expressions (composition of polynomials, division and taking roots). Using Gröbner basis, it is possible to construct an algorithm that computes results (now, I'm not sure if it's always possible, I'm not an expert. But this distinction is important nonetheless).

[ransac9000]

If you can't algebraically solve the univariate polynomial, you can still get some solutions to the system as long as you can obtain some solutions to the univariate polynomial numerically.