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Solving polynomials is HARD. Here's why.
Use my code, 2SWAPYT20, for a 20% discount on your first purchase of a Lovable Pro plan before December 26!
Click here to automatically apply the discount at checkout: https://lovable.link/2swapyt
Check out the interactive polynomial solutions demo here: https://2swap.github.io/LittlewoodFractal/
If you liked this, please support me on Patreon! https://www.patreon.com/2swap
You can support the musician, 6884, on Kofi: https://ko-fi.com/iam6884
Check out his music here! https://6884.bandcamp.com/
This video was animated using SwapTube! https://github.com/2swap/swaptube
Join our discord server here, where we talk about math, CS, puzzles, etc! https://discord.gg/XDb7x2BZAE
=== Nerd Notes ===
There are a number of comments asking "But what if I use [something] in my quintic formula?" One such [something] being the Bring radical, which I will take as an example here. The Bring radical is defined as the solution of a particular quintic, which can then be used to solve quintics in the general case. (Aside: I'm not sure this can be called a formula because even by invoking the Bring radical in the general case requires casework, per all of the sources I have found. If you know of one which does not, leave a comment.)
But as far as I am concerned, this yields a larger question of what can be called a formula in the first place. If I say that an oracle which yields the solutions of the quintic is a valid symbol in my "formula", then the formula is merely the identity statement that the solution to the quintic is the solution to the quintic. I think this sidesteps the question- and furthermore, in such a world, the halting problem would even be solvable, by merely invoking such an oracle. The idea of a formula, if you ask me, is that we represent a complex behavior with simpler constituent symbols which we accept as atomic computations. Where the line is drawn is, of course, arbitrary, (although there is certainly interesting things to be learned by accepting alternate symbols,) but in this video we consider the arithmetic operators and traditional radicals.
=== Errata ===
Found by @samuelwaid: at 12:26 I said "we define i as being the [square root of -1]" but the actual convention is to define i as being an element such that i^2 = -1. As mentioned in the video, there are various valid definitions of the square root function, and for some of them, the "definition" of i mentioned in the video would not be respected.
Found by @fumeal: at 11:37 I said that the set of operators on screen were continuous and single valued. However the division operator is not continuous at z=0.
Found by myself: in the conclusion I said that there is no solution writable in terms of algebraic functions. This isn't at all correct, since the solution itself is an algebraic function! So, one may just say that the solution of the quintic is the solution of the quintic, and be done with it. For some reason I thought that algebraic functions were the functions definable with radicals and traditional operators, which is not the case (and obviously so, by juxtaposition with the definition of the algebraic numbers.) Sigh. If you replace "algebraic" with "using radicals and arithmetic operators", this would be true.
Found by many: the fundamental theorem of Algebra does not comment on constant polynomials, which I didn't mention in the video.
=== Sources and References ===
This video was primarily inspired by a paper by Leo Goldmakher. It covers the main proof of this video in a more academic fashion. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf
The proof itself is by Vladimir Arnold: https://en.wikipedia.org/wiki/Vladimir_Arnold
While I was interested in Leo's paper, I saw this post about fractals of polynomial solutions. The images were somewhat low-resolution, which inspired me to make my own in conjunction with the topic of the video. https://les-mathematiques.net/vanilla/discussion/2335252/images-des-entiers-algebriques / https://web.archive.org/web/20240530132413/https://les-mathematiques.net/vanilla/discussion/2335252/images-des-entiers-algebriques
This excellent talk by Andrej Bauer was recommended to me while making my video. It goes much deeper into detail about the fractals of roots seen here. His talk prompted the comment about the dragon curve in this video. https://www.youtube.com/watch?v=8wrYPqEU1x0
Here are some other resources about the Abel-Ruffini theorem which I found along the way that helped with my own understanding:
This tool helped me enormously in animating the paths of my roots and coefficients through complex space. Note the box that shows the roots of the determinant- compare to the "poles" in my animation: https://duetosymmetry.com/tool/polynomial-roots-toy/
Another easy-to-read paper covering this proof: https://arxiv.org/abs/2011.05162
Two more youtube videos covering this proof: https://www.youtube.com/watch?v=BSHv9Elk1MU https://www.youtube.com/watch?v=RhpVSV6iCko
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By Use my code, 2SWAPYT20, for a 20% discount on your first purchase of a Lovable Pro plan before December 26!
Click here to automatically apply the discount at checkout: https://lovable.link/2swapyt
Check out the interactive polynomial solutions demo here: https://2swap.github.io/LittlewoodFractal/
If you liked this, please support me on Patreon! https://www.patreon.com/2swap
You can support the musician, 6884, on Kofi: https://ko-fi.com/iam6884
Check out his music here! https://6884.bandcamp.com/
This video was animated using SwapTube! https://github.com/2swap/swaptube
Join our discord server here, where we talk about math, CS, puzzles, etc! https://discord.gg/XDb7x2BZAE
=== Nerd Notes ===
There are a number of comments asking "But what if I use [something] in my quintic formula?" One such [something] being the Bring radical, which I will take as an example here. The Bring radical is defined as the solution of a particular quintic, which can then be used to solve quintics in the general case. (Aside: I'm not sure this can be called a formula because even by invoking the Bring radical in the general case requires casework, per all of the sources I have found. If you know of one which does not, leave a comment.)
But as far as I am concerned, this yields a larger question of what can be called a formula in the first place. If I say that an oracle which yields the solutions of the quintic is a valid symbol in my "formula", then the formula is merely the identity statement that the solution to the quintic is the solution to the quintic. I think this sidesteps the question- and furthermore, in such a world, the halting problem would even be solvable, by merely invoking such an oracle. The idea of a formula, if you ask me, is that we represent a complex behavior with simpler constituent symbols which we accept as atomic computations. Where the line is drawn is, of course, arbitrary, (although there is certainly interesting things to be learned by accepting alternate symbols,) but in this video we consider the arithmetic operators and traditional radicals.
=== Errata ===
Found by @samuelwaid: at 12:26 I said "we define i as being the [square root of -1]" but the actual convention is to define i as being an element such that i^2 = -1. As mentioned in the video, there are various valid definitions of the square root function, and for some of them, the "definition" of i mentioned in the video would not be respected.
Found by @fumeal: at 11:37 I said that the set of operators on screen were continuous and single valued. However the division operator is not continuous at z=0.
Found by myself: in the conclusion I said that there is no solution writable in terms of algebraic functions. This isn't at all correct, since the solution itself is an algebraic function! So, one may just say that the solution of the quintic is the solution of the quintic, and be done with it. For some reason I thought that algebraic functions were the functions definable with radicals and traditional operators, which is not the case (and obviously so, by juxtaposition with the definition of the algebraic numbers.) Sigh. If you replace "algebraic" with "using radicals and arithmetic operators", this would be true.
Found by many: the fundamental theorem of Algebra does not comment on constant polynomials, which I didn't mention in the video.
=== Sources and References ===
This video was primarily inspired by a paper by Leo Goldmakher. It covers the main proof of this video in a more academic fashion. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf
The proof itself is by Vladimir Arnold: https://en.wikipedia.org/wiki/Vladimir_Arnold
While I was interested in Leo's paper, I saw this post about fractals of polynomial solutions. The images were somewhat low-resolution, which inspired me to make my own in conjunction with the topic of the video. https://les-mathematiques.net/vanilla/discussion/2335252/images-des-entiers-algebriques / https://web.archive.org/web/20240530132413/https://les-mathematiques.net/vanilla/discussion/2335252/images-des-entiers-algebriques
This excellent talk by Andrej Bauer was recommended to me while making my video. It goes much deeper into detail about the fractals of roots seen here. His talk prompted the comment about the dragon curve in this video. https://www.youtube.com/watch?v=8wrYPqEU1x0
Here are some other resources about the Abel-Ruffini theorem which I found along the way that helped with my own understanding:
This tool helped me enormously in animating the paths of my roots and coefficients through complex space. Note the box that shows the roots of the determinant- compare to the "poles" in my animation: https://duetosymmetry.com/tool/polynomial-roots-toy/
Another easy-to-read paper covering this proof: https://arxiv.org/abs/2011.05162
Two more youtube videos covering this proof: https://www.youtube.com/watch?v=BSHv9Elk1MU https://www.youtube.com/watch?v=RhpVSV6iCko