/sci/ - Science & Math

[Anonymous No.15070175]

Can /sci/ explain in layman terms what a Cox-Zucker machine is used for?

[Anonymous No.15070213]

lmaoooooooo

[Anonymous No.15070266]

They're called women, anon

[Anonymous No.15070309]

>>15070175
It's a way of computing the solutions to an extremely specific and arguably boring set of component parameter of an arguably even more specific and boring shape surface

[Anonymous No.15070572]

>it's real
kek

[Anonymous No.15070612]

>>15070175
>linux font rendering faggot

[Anonymous No.15070617]

>>15070175
We thank them for their service

[Anonymous No.15070628]

>>15070175
They did it for the memes.

[Anonymous No.15071203]

>>15070572
LMAO

[Anonymous No.15071819]

>>15070175
Sure, so first you need to know a little about elliptic curve: you can think of an elliptic curve [math]E[/math] defined over [math]\mathbb{Q}[/math] (we're working over [math]\mathbb{Q}[/math] to keep it simple) as the zero locus of a polynomial of the form [math]y^2-x^3+ax+b[/math], with [math]a,b[/math] rationals (I know it sounds a bit too specific when you hear it the first time, but there are good reasons why such curves are interesting). Now a bit of a caveat, to make things run more smoothly we need to "complete" this curve by adding a point at infinity, that we call O. Again, there are very good reasons for this
Of course, you can try to find a zero [math]P=(x,y)[/math] with rational coefficients, and we call [math]E( \mathbb{Q})[/math] the set of all such points. This set is actually an abelian group, its addition is cute from a geometric standpoint but ugly from an algebraic standpoint. You can find the details on wikipedia or anywhere really. We call [math]E( \mathbb{Q})[/math] the Mordell-Weil group.
An important result about this group is the Mordell-Weil theorem saying that it is finitely generated.

Now, onto families of varieties: what does wikipedia mean by "elliptic surface [math]E \to S[/math]"? It means that [math]S[/math] is the projective line [math]\mathbb{P}^1[/math], it's a line with an added point at infinity; and basically, for each [math]s \in S[/math], you can take the fiber [math]E_s[/math] (think the set-theoretic inverse image of [math]s[/math]), and we require that this fiber be an elliptic curve, for each [math]s[/math].
In laymen's terms, we are provided with a family of elliptic curves [math]\{ E_s \}[/math] indexed by [math]s \in \mathbb{P}^1[/math].

Now as we know, for [math]s[/math] fixed we have [math]E_s(\mathbb{Q})[/math] finitely generated, so we may find what wikipedia calls a basis up to torsion, say [math]P_1(s),\dots , P_r(s)[/math]. Now the real challenge lies in "patching" these choices of point together

[Anonymous No.15071840]

>>15071819
By which I mean: when we let [math]s[/math] vary, will my point [math]P_1(s)\in E[/math] vary "nicely" (think continuously)? When this happens we say that [math]P_1(s)[/math] is a section of [math]E\to S[/math].
Of course we may ask the same of [math]P_2,\dots, P_r[/math].
If we manage to choose a basis up to torsion [math]P_i(s)[/math] for each [math]s[/math] in such a way as to make each [math]P_i(s)[/math] a section, then we call these [math]P_1(s),\dots, P_r(s)[/math] a basis up to torsion of my elliptic surface.

Now this turns out to be a very difficult question to answer (we actually don't even know how to find explicit [math]P_1,\dots, P_r[/math] for fixed [math]s[/math]!), but the Cox-Zucker machine does something more humble: if we feed some sections [math]P_1(s),\dots,P_r(s)[/math] into it, it will tell us whether they actually get the job done, i.e. it checks whether they actually are a basis up to torsion, and algorithmically at that

[Anonymous No.15071850]

>>15071819
I just noticed a typo, the polynomial should be [math]y^2-x^3-ax-b[/math]