875: Structural Proof Theory/7

Buzzard, Kevin M k.buzzard at imperial.ac.uk
Sun Mar 21 06:08:23 EDT 2021

Re: Tiny Diophantine equations. There have been thousands of years of research
into Diophantine equations and one thing we have learnt is a very good
notion of "order of difficulty". For example a general degree two equation in 1000 variables
is much easier than a general degree 3 equation in 2 variables. The discussion
about Diophantine equations is sufficiently vague for it to be very difficult
to say anything coherent. The moment you allow for high degree you
instantly run into all sorts of unsolved problems, even in very simple
cases. This foundational way of measuring "tiny" as some way of describing
how many characters one needs to fit a description of your equation into
your theory is an extremely poor one when it comes to actual Diophantine
complexity, which is governed by degree and dimension of the associated
algebraic variety (the set of complex solutions to the equation), by profound
theorems relating geometry and arithmetic such as Faltings' theorem.
This is why x^3-2y^3=1 is profoundly more difficult than a degree two
equation in 100 variables. Already there are issues when moving from
degee 2 to degree 3; for example there is currently no known algorithm
for deciding whether a given degree three equation in two variables
has a rational solution (or equivalently, clearing denominators, whether
a given degree 3 equation in 3 variables has a non-zero integer solution).
However in all cases where the coefficients are small, people have used
tricks to figure this out; what is not known is whether the tricks always work.
This is to do with something called the finiteness of the Tate-Shaferevich
group of an elliptic curve, a famous open problem (it is conjectured to be always
finite, meaning that the algorithm we have will always terminate). For
me, Harvey's unspecified model of what it means for a Diophantine equation to
be TINY has too few details of what TINY actually means to be able to make
a sensible response. For example can we talk about degree 3 equations in two
variables? If so then one is already in a lot of trouble. It would not surprise
me if existence of natural number solutions to x^3-5y^4=1 is an open problem
(and if it's not then just tinker with -5 and 1 a little until it is). Is that tiny?


I started a project some years back on TINY DIOPHANTINE EQUATIONS...
Harvey Friedman
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