What are...Gröbner bases?

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.

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.

Reduction.

Some applications.

How to plot varieties.

Mathematica.

[pedrogonzalez]

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

[AlfonsoMA]

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

[MuhammadAbraarAbhirama]

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 ;)

[larubiano]

Pretty interesting, thanks for the video

[davidstork]

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.

[TheKivifreak]

Thanks, now I can solve multivariate polynomials more easily.

[carstenjantzen]

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

[mattiapiras]

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...

[davidstork]

Is the Mathematica demonstration available?

[wacharinwichiramala]

Typos? Should be s=red(s(p, q), G') and G=GU{s}?

[subnow]

Where have Grobner bases been all my life?!