Apologies for the vague title. Part of my question is what the technically accurate language is for this function $f$ I'm asking about.
- The Function
I found this map on $\mathbb C^3$ I'd like to learn more about. It takes a triplet of numbers $(x,y,z)$ and produces a new triplet using three rational functions.
$$ \small x\overset{f}{\longmapsto} \frac {x^2yz + 2xy^2z + y^3z + 2xyz^2 + 2y^2z^2 + yz^3 - x^3 - xy^2 - 2x^2z - xz^2 + 2x^2 - 2xy + 2xz - 3yz - x - 2z} {x^2yz + 2xy^2z + y^3z + 2xyz^2 + 2y^2z^2 + yz^3 - x^2y + y^3 + x^2z + y^2z + 2xz^2 + yz^2 + z^3 - x^2 + 2xy + y^2 - 2xz + yz - z^2 + 2x - y + z - 1} $$
$$ \small y\overset{f}{\longmapsto} \frac {-x^3z + xy^2z - 2x^2z^2 - xz^3 - x^2y - y^3 - 2y^2z - yz^2 - 2y^2 + xz - 4yz - y - 2z} {x^3z + 2x^2yz + xy^2z + 2x^2z^2 + 2xyz^2 + xz^3 - x^3 + xy^2 - x^2z - y^2z - xz^2 - 2yz^2 - z^3 + x^2 + 2xy - y^2 + xz - 2yz - z^2 + x - 2y - z - 1} $$
$$ \small z\overset{f}{\longmapsto} \frac {x^3y - xy^3 + 2x^2yz + xyz^2 - 2x^2y - 2xy^2 - x^2z - 4xyz - y^2z - 2xz^2 - 2yz^2 - z^3 - xy - 2yz - z} {x^3y + 2x^2y^2 + xy^3 + 2x^2yz + 2xy^2z + xyz^2 + x^3 + x^2y - xy^2 - y^3 + 2x^2z - 2y^2z + xz^2 - yz^2 - x^2 - 3xy - y^2 - 2xz - 2yz - z^2 - x + y + 1} $$
- Weird Properties
This map has two weird properties.
- The map $f$ is an involution (i.e. its own inverse). So $f(f(x,y,z))=(x,y,z)$. This property felt like a miracle so I coded the whole thing up on SageMath to be sure.
x, y, z = var('x y z')
f_x = x^2*y*z + 2*x*y^2*z + y^3*z + 2*x*y*z^2 + 2*y^2*z^2 + y*z^3 - x^3 - x*y^2 - 2*x^2*z - x*z^2 + 2*x^2 - 2*x*y + 2*x*z - 3*y*z - x - 2*z
f_x /= x^2*y*z + 2*x*y^2*z + y^3*z + 2*x*y*z^2 + 2*y^2*z^2 + y*z^3 - x^2*y + y^3 + x^2*z + y^2*z + 2*x*z^2 + y*z^2 + z^3 - x^2 + 2*x*y + y^2 - 2*x*z + y*z - z^2 + 2*x - y + z - 1
f_x = f_x.full_simplify()
print('f_x =', f_x)
print(' ')
f_y = -x^3*z + x*y^2*z - 2*x^2*z^2 - x*z^3 - x^2*y - y^3 - 2*y^2*z - y*z^2 - 2*y^2 + x*z - 4*y*z - y - 2*z
f_y /= x^3*z + 2*x^2*y*z + x*y^2*z + 2*x^2*z^2 + 2*x*y*z^2 + x*z^3 - x^3 + x*y^2 - x^2*z - y^2*z - x*z^2 - 2*y*z^2 - z^3 + x^2 + 2*x*y - y^2 + x*z - 2*y*z - z^2 + x - 2*y - z - 1
f_y = f_y.full_simplify()
print('f_y =', f_y)
print(' ')
f_z = x^3*y - x*y^3 + 2*x^2*y*z + x*y*z^2 - 2*x^2*y - 2*x*y^2 - x^2*z - 4*x*y*z - y^2*z - 2*x*z^2 - 2*y*z^2 - z^3 - x*y - 2*y*z - z
f_z /= x^3*y + 2*x^2*y^2 + x*y^3 + 2*x^2*y*z + 2*x*y^2*z + x*y*z^2 + x^3 + x^2*y - x*y^2 - y^3 + 2*x^2*z - 2*y^2*z + x*z^2 - y*z^2 - x^2 - 3*x*y - y^2 - 2*x*z - 2*y*z - z^2 - x + y + 1
f_z = f_z.full_simplify()
print('f_z =', f_z)
print(' ')
ff_x = f_x(x=f_x, y=f_y, z=f_z).full_simplify()
ff_y = f_y(x=f_x, y=f_y, z=f_z).full_simplify()
ff_z = f_z(x=f_x, y=f_y, z=f_z).full_simplify()
print(ff_x)
print(ff_y)
print(ff_z)
Assuming the code is correct, the involution checks out.
- The function $f$ maps the set of "Flipping Fractions" (as I'll call them) into itself. An example of "Flipping Fractions" is $$\frac{7}{10}+\frac{7}{4}+\frac{4}{5}=\frac{5}{4}+\frac{4}{7}+\frac{10}{7}$$ I.e. a sum of three fractions that's fixed by "180 degree rotation". I found this map $f$ when trying to understand these "Flipping Fractions" triplets.
For an example of this second property: $$ \Big(\frac{7}{10},\frac{7}{4},\frac{4}{5}\Big) \overset{f}{\longmapsto} \Big(\frac{50}{341},\frac{275}{62},\frac{31}{11}\Big) $$ And indeed, the new triplet $$\frac{50}{341}+\frac{62}{275}+\frac{31}{11}=\frac{11}{31}+\frac{275}{62}+\frac{341}{50}$$ is in fact an instance of "Flipping Fractions".
- Language Questions
I've been learning algebraic geometry and wanted firstly to ask what the correct label would be for this map?
My thinking is that it's a rational map with rational inverse, hence a "birational map" from $\mathbb A_{\mathbb C}^3$ to itself, and hence a "birational involution on affine $3$-space". The set of Flipping Fractions is a subset of the variety $$X=V(E_1E_3-E_2)$$ where $E_j$ is the symmetric polynomial of degree $j$ in $3$ variables since $$E_1=x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{xy+zx+yz}{xyz}=\frac{E_2}{E_3}$$ So it seems like I'd want to say that this "birational involution" restricts to an "birational automorphism of the quartic surface $X$".
I'm mostly just curious which terms I should be looking up to find relevant AG results for this function. The only other similar functions that ocurred to me when researching this are Mobius Transformations, i.e. the linear automorphisms of $\mathbb P_{\mathbb C}^1$, since this function $f$ is the restriction of a rational automorphism of $\mathbb P_{\mathbb C}^3$. Or well, I think so at least. My thinking is that this map $f$ would extend uniquely from $\mathbb A_{\mathbb C}^3$ to $\mathbb P_{\mathbb C}^3$, and that the extension would correspond to the variety $$ Y=V(E_1E_3-E_2w^2)$$ where the variable $w$ has been added to homogenize.
Anyways, this other question I found while writing this up feels like a validation of the comparison to Mobius Transformations: Involution on triplets
- Math Questions
So the main thing I'm wondering is whether this map tells me anything about "Flipping Fractions". Like how many there are, whether any "Flipping Fractions" exist with all numerators/denominators perfect squares, whether we can give a deterministic routine to list all "Flipping Fractions". The standard questions about rational points I think.
It feels like I'd want a generalization of the Mordell-Weil group at this point. Like, I'm curious if this involution (along with the "obvious" automorphisms) can generate every rational point of $Y$ from some finite initial set of points. I.e. does this map give us some way to present the rational points of $Y$ (denoted $\mathbb{Q}(Y)$?) as a finitely generated group?
I'm also wondering if involutions like this are rare? It feels like a big jump going from Mobius Transformations to this beast. It makes me wonder if there isn't something simpler in between. Or for that matter, whether is there some simple and explanatory relationship of this map $f$ to say $PGL(6, \mathbb C)$ or a subgroup of $PGL(2, \mathbb H)$. The smell of it reminds me of the enumeration of 3D rotations by quaternion bi-multiplication for some reason.
Is there a simpler birational involution on $\mathbb A_{\mathbb C}^3$? Well, besides the "obvious" ones like $$(x,y,z)\longmapsto (-x, -y, -z)$$ And for that matter, is there a decent way to visualize what this map $f$ is doing?
Any help is appreciated.
