I got an email the other day from mathematician Bogdan Grechuk, whose book Polynomial Diophantine Equations: A Systematic Approach (https://doi.org/10.1007/978-3-031-62949-5) was recently released. This is to my mind a rather remarkable book. The blurb starts:
This book proposes a novel approach to the study of Diophantine equations: define an appropriate version of the equation’s size, order all polynomial Diophantine equations by size, and then solve the equations in order. [emphasis added]
The book really does do this systematically. The start of the book solves extremely easy examples, but pays close attention to the process, so that it can identify classes of equations that this method will apply to. And as a new method of solution is provided, hence solving a new class of equations, this class of equations is removed from considerations going forward. Grechuk, halfway through the book (>400 pages in!) takes stock and provides an outline of the algorithm built up over the four chapters to that point. It has approximately 21 main cases, of increasing difficulty, spread over pages, referring constantly back through the book to various methods and algorithms for different cases. As the equations keep getting harder, the problem of solution shifts focus: from “describe all solutions parametrically”, to “write down a family of polynomials (or rational functions) that give all solutions”, to “find a recurrence relation to describe solutions”, to “determine if there are finitely or infinitely many solutions”, to “determine if there is any one solution”. The methods start out wholly elementary, and by the end the discussion uses the cutting edge of current techniques for attacking number-theoretic problems of this sort, for instance Chabauty methods.
Long-time readers of the blog may recall the post Diophantine fruit. which was inspired by a MathOverflow question of Grechuk on the problem of “the smallest unsolved Diophantine equation”. This blog post led to a chain of papers using the phrase “fruit equations” by authors wholly unconnected with me, with the first solving what was at the time an open problem on Grechuk’s list. So it’s worth consulting the companion paper A systematic approach to Diophantine equations: open problems (https://arxiv.org/abs/2404.08518) if you have any particular hankering to solve an equation (in any of the various ways: from parametrising a family of solutions to just finding one single solution) and have the honour of being the first person ever (modulo the problem of finding some equation equivalent to it buried in the literature) to solve a hard equation. The book ends with a summary that is a line-in-the-sand version of the just-cited arXiv preprint, stating the earliest/smallest/shortest equations that the various types of solutions are not known. And the last open problem is this:
What is the smallest (in H) equation for which the existence of integer solutions (Problem 7.1) is a problem which is independent from the standard axioms of mathematics (Zermelo–Fraenkel set theory with the axiom of choice)?
Here “H” refers to the function that gives the “size” of a polynomial Diophantine equation, that allows a systematic ordering. Other natural orderings are given in the book, which end up being more-or-less comparable, if not the same, and the above problem is also posed for these orderings. From the MDRP solution to Hilbert’s Tenth Problem we know that at some point equations whose solvability is unprovable in ZFC will turn up. By work of Zhi-Wei Sun (https://arxiv.org/abs/1704.03504) we know that the unsolvability result (that is, there is no algorithm that can solve in integers all equations in a given class) is true taking even just Diophantine equations with no more than 11 variables. But identifying an explicit equation, let alone the smallest one, seems very hard. Moreover, trying to optimise to find a small ZFC-undecidable equation, rather than an algorithmically unsolvable one, is another whole kettle of fish; compare how the value BB(745) of the Busy Beaver function is not possible to calculate in ZFC, through a line of work whose current endpoint is by Johannes Riebel’s 2023 Bachelor thesis. (ADDED: I just now found that in fact BB(636) is known to uncalculable in ZFC, by very recent work of Rohan Ridenour in the past two months)
Just as Grechuk’s book starts small and tries to find the smallest so-far unsolved Diophantine equations, here is an example of one such equation, but one that is far from small:
y^2 + xy = x^3 - 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497(source: https://web.math.pmf.unizg.hr/~duje/tors/rk29.html). This equation defines an elliptic curve, and it has the largest-known number of solutions in rational numbers of any such equation. By “largest-known” I mean that there is a copy of in the set of rational solutions (where we are in fact finding solutions in the projective plane, not just in affine coordinates as presented). That is, there are 29 rational solutions (shown at the previous link) that are linearly-idependent (over the integers) under the abelian group operation on the rational points. It is not proved that there are no other linearly independent solutions (this number is known as the rank of the elliptic curve described by the equation). The announcement of the result as well as a summary of how much more is known is given in the email:
The above elliptic curve has been proved to have rank exactly 29 using the Generalised Riemann Hypothesis for zeta functions of number fields, which is very far from being known (the usual Riemann Hypothesis is the special case of taking the number field to be the rationals). So the specific two-variable cubic equation above is an example of a polynomial Diophantine equation whose complete solution—and I haven’t even mentioned the possible (finite) torsion subgroup of the elliptic curve—requires knowing a the resolution of a conjecture that is wholly out of reach of current mathematics.
(source for image: Mark Hughes, How Elliptic Curve Cryptography Works, 2019)
https://thehighergeometer.wordpress.com/2024/09/08/two-items-a-new-rare-high-rank-elliptic-curve-and-a-beautifully-organised-orchard-of-diophantine-equations/
#DiophantineEquations #Elkies #ellipticCurves #Grechuk #Klagsbrun
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