What are...Gröbner bases?

Рет қаралды 7,472

3 жыл бұрын

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What are...Gröbner bases? Or: Minimal intersections.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Typos (sorry).
As pointing out in the comments, in Buchberger's algorithm it should be "s=red(s(p,q),G') and G=GU{s}".
Slides.
www.dtubbenhauer.com/youtube.html
Thumbnail equations.
(X*Z + Y^2 - 1=0, X*Z - Y^2 - X*Y=0) and (-1 + X Y + 2 Y^2=0, -1 - X Y + 2 X Z=0)
Material used.
en.wikipedia.org/wiki/Gr%C3%B...
de.wikipedia.org/wiki/Buchber...
mattpap.github.io/masters-the...
Reduction.
en.wikipedia.org/wiki/Gr%C3%B...
Some applications.
www.msri.org/people/members/c...
How to plot varieties.
www.singsurf.org/index.php
pi.math.cornell.edu/~dmehrle/n...
Mathematica.
demonstrations.wolfram.com/Th...
reference.wolfram.com/languag...
reference.wolfram.com/languag...

Пікірлер: 26

@pedrogonzalez8873 2 жыл бұрын

Very visual, very clear explanation of a beautiful topic. Thanks a lot, now I made myself clear about the minimal intersection between ideals.💡

@VisualMath 2 жыл бұрын

Visuals are promised, of course ;-) But I am always glad to hear that other people feel the same about the exposition as I do. I try to explain a topic the way it works for me. Its very important feedback to hear that I am not the only one for whom the explanation works, thank you so much. And I really do not deserve your praise since, yup, the topic is beautiful, which made my life easy ;-)

@MuhammadAbraarAbhirama Ай бұрын

This is beyond great, unlike other videos that are not straight to the main idea! Would you like to make some videos about rings of differential operators, particularly with polynomial coefficients? It is highly related to Gröbner Bases (and of course, Weyl Algebra). I am currently studying it for my thesis. Thank you! Also, I have already hit that subscribe and like button ;)

@VisualMath Ай бұрын

Thanks for watching 😀 I guess you are studying some form of algebraic geometry? At the moment I am not planing anything on the Weyl algebra, but we will see what the future holds.

@Alfonso00MA Жыл бұрын

Great explanation. I'm looking forward to see some of your other videos in your playlist. Thanks

@VisualMath Жыл бұрын

Awesome, thank you! I hope you enjoy your math journey and that some videos will turn out helpful!

@alexfotue 2 жыл бұрын

Thanks to this course.

@VisualMath 2 жыл бұрын

Glad that you liked it. I hope it was useful.

@larubiano0 3 ай бұрын

Pretty interesting, thanks for the video

@VisualMath 3 ай бұрын

I am glad that you liked the Groebner bases 😚

@mattiapiras3331 Жыл бұрын

Nice video and great explanation. Sorry in advance for my questions that can be trivial . I understand the enormous implications this can have but what are the limitations of this approach ? Do you think there could be the possibility to use interval analysis with this algorithm or the whole story would drastically change in that case? Thanks in advance...

@VisualMath Жыл бұрын

Excellent question! - The main problem with the Gröbner approach is it rather bad run-time. The worst-case complexity to compute Gröbner bases is about d^(2^n) (double exponential!) where n=number of variables, and d=maximal degree of the involved polynomials. Gröbner bases can often be computed faster (Gröbner bases computation are useful beyond toy examples!), but a double exponential worst case scenario is a problem. - To come to your final question, Gröbner bases make sense in quite some generality. For example, noncommutative or tropical versions are quite popular. I haven’t seen anyone using it for interval calculus. However, tropical arithmetic is, let us say, very strange and Gröbner bases are still very useful here. So why not, I think that would be nice to explore.

@mattiapiras3331 Жыл бұрын

@@VisualMath thank you very much. I have really appreciated your feedback

@VisualMath Жыл бұрын

@@mattiapiras3331 Welcome!

@carstenjantzen1341 2 жыл бұрын

Thanks a lot. Göbner seems to very easy... after awhile

@VisualMath 2 жыл бұрын

I am glad that you liked it. And indeed, Gröbner is very easy after a while, agreed! Great ideas are always easy, aren't they? ;-) It is definitely worthwhile to spend time talking about them and learning what they are about. Of course, we are all humans, so it takes a while to digest ideas. It took me ages to understand Gröbner bases up to a point I can write "I understand them" without feeling stupid. (There is of course still a lot to discover about them!) But they are so cute in hindsight! I hope the video was reasonably helpful to get some hints what Gröbner bases are all about. Gröbner bases are so easy that one can feed the algorithm into a machine and any reasonable CAS should run it in the background. Its such a great idea, love it! I subscribe to Gröbner bases ;-)

@davidstork5604 4 ай бұрын

Is the Mathematica demonstration available?

@VisualMath 4 ай бұрын

Did I forgot to link it? My bad 😥, use this one here: demonstrations.wolfram.com/TheBuchbergerGroebnerBasisAlgorithm/

@TheKivifreak Жыл бұрын

Thanks, now I can solve multivariate polynomials more easily.

@VisualMath Жыл бұрын

Glad I could help! Gröbner bases are pretty cool and applicable (which is a rare combination), so I hope you enjoy working with them.

@davidstork5604 4 ай бұрын

Very helpful. Alas, your demonstration software (Mathematica demonstration) never seems to include cross terms, e.g., x y. Also, it would be great if it could accept ideals from users.

@VisualMath 4 ай бұрын

Thanks for the suggestion: I do not know any program which illustrates this nicely and is user friendly, so this might be a good project for the future.