What is...the Ax-Grothendieck theorem?

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 is...the Ax-Grothendieck theorem? Or: Polynomials rule!

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.
http://www.dtubbenhauer.com/youtube.html

Thumbnail.
Pictures created Mathematica’s plot for example https://reference.wolfram.com/language/ref/Plot.html

Main discussion.
https://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem
https://math.osu.edu/sites/math.osu.edu/files/AxGrothendieck.pdf
https://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/
Section 3 of https://arxiv.org/abs/0903.0517
https://golem.ph.utexas.edu/category/2012/09/the_axgrothendieck_theorem_acc.html
https://proofwiki.org/wiki/Ax-Grothendieck_Theorem
https://xorshammer.com/2008/08/15/axs-theorem/
https://modeltheory.fandom.com/wiki/Ax-Grothendieck_Theorem
https://www.jstor.org/stable/2034464

Background.
https://en.wikipedia.org/wiki/Polynomial
https://en.wikipedia.org/wiki/Injective_function
https://en.wikipedia.org/wiki/Surjective_function
https://en.wikipedia.org/wiki/Bijection
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
https://en.wikipedia.org/wiki/Complex_number
https://en.wikipedia.org/wiki/Finite_field
https://en.wikipedia.org/wiki/Algebraically_closed_field
https://en.wikipedia.org/wiki/Algebraic_variety
https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
https://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
https://en.wikipedia.org/wiki/Garden_of_Eden_(cellular_automaton)

Mathematica.
https://demonstrations.wolfram.com/topic.html?limit=20&topic=Polynomials

Pictures used.
Pictures created Mathematica’s plot for example https://reference.wolfram.com/language/ref/Plot.html see also https://reference.wolfram.com/language/ref/Plot3D.html and related plotting tools
Picture created using Mathematica’s complex plot for example https://reference.wolfram.com/language/guide/ComplexVisualization.html see also https://www.wolfram.com/language/12/complex-visualization/complexplot-introduction.html?product=mathematica
https://m.media-amazon.com/images/I/41sjL0WwGuL.jpg

YouTube and co.
https://www.youtube.com/watch?v=3ar_H4Z8Wbk
https://www.youtube.com/watch?v=ZaT_ciXoc40
https://www.youtube.com/watch?v=shEk8sz1oOw

#algebra
#algebraicgeometry
#mathematics

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