Importance of determining whether a number is squarefree, using geometry

Question

Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important unsolved problems in arithmetic geometry. He told me about a few, but along with some well-known problems he told me the following one also:

How to determine conceptually when a number is squarefree or not?

When I protested that this sounded like a computational question, he told me that no, this is not so, and demonstrated that this has a rather nice solution for the ring of polynomials over a field, which is in many senses analogous to the ring of integers. Take the polynomial, take its derivative and compute the gcd using the Euclidean algorithm. But for the ring of integers there is nothing analogous to the derivative, and he wanted a solution of the problem by constructing a good notion of a differential in this case.

Question: What are the known investigations along this line? What well-known topics in arithmetic geometry are related to this? And what would be some other interesting consequences of a successful development of such a method?

Any other comments that might enlighten me further would be received with gratitude.

Answer 1

The standard example is the proof, by differentiation, of Fermat's last theorem for polynomials. More generally, the proof of the ABC conjecture for polynomials (Mason's theorem) by differentiation. In light of the analogies between algebraic number theory and algebraic geometry this suggests the hope that some kind of arithmetic differentiation, if it exists, could be a missing structure leading to breakthroughs such as a proof of the ABC conjecture or a simple proof of Fermat.

As somebody once said, "define PDE's over a number field and you'll be a rich man".

As far as I know the furthest-developed approach to this problem at present is in a series of works by Alexander Buium on arithmetic differential operators:

http://www.math.unm.edu/~buium/prebook.pdf

Answer 2

The problem of finding some sort of analogue of differentiation of polynomials, in the context of integers, is a celebrated and difficult one. I don't have anything useful to say about it, except to remark that the standard context in which this problem arises is that of the ABC conjecture: namely, the polynomial analogue of the ABC conjecture is a theorem, and one of the tools in its proof involves differentiating polynomials. This has led people to try to construct some sort of analogous tool over the integers, with the goal of proving the ABC conjecture.

There is a short article of Faltings (called "Does there exist an arithmetic Kodaira--Spencer class") in which he writes down some reasonable axioms such a notion of differentiation would satisfy (in scheme theoretic language over Spec $\mathbb Z$) and shows that it is impossible to satisfy them. Nevertheless, one might (and many do!) hope for some more subtle structure, more complicated than the direct analogue that Faltings rules out, but still having the same basic nature as a derivative.

[Added September 5 2012:] Mochizuki has just released a series of papers, building on his earlier work, claiming to prove ABC.

Answer 3

No feasible (polynomial time) algorithm is currently known for recognizing squarefree integers or for computing the squarefree part of an integer. In fact it may be the case that this problem is no easier than the general problem of integer factorization.

This problem is important because one of the main tasks of computational algebraic number theory reduces to it (in deterministic polynomial time). Namely the problem of computing the ring of integers of an algebraic number field depends upon the square-free decomposition of the polynomial discriminant when computing an integral basis, e.g. [2] S.7.3 p.429 or [1] This is due to Chistov [0]. See also Problems 7,8, p.9 in [3], which lists 36 open problems in number theoretic complexity.

The reason that such problems are simpler in function fields (e.g. polynomial rings) versus number fields is due to the availability of derivatives. This opens up a powerful toolbox that is not available in the number field case. For example once derivatives are available so are Wronskians - which provide powerful measures of dependence in transcendence theory and diophantine approximation. A simple yet stunning example is the elementary proof of the polynomial case of Mason's ABC theorem, which yields as a very special case FLT for polynomials, cf. my recent MO post and my old sci.math post [4].

[0] A. L. Chistov. The complexity of constructing the ring of integers
of a global field. Dokl. Akad. Nauk. SSSR, 306:1063--1067, 1989.
English Translation: Soviet Math. Dokl., 39:597--600, 1989. 90g:11170
http://citeseerx.ist.psu.edu/showciting?cid=854849

[1] Lenstra, H. W., Jr. Algorithms in algebraic number theory.
Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 211--244. 93g:11131
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.105.8382

[2] Pohst, M.; Zassenhaus, H. Algorithmic algebraic number theory.
Cambridge University Press, Cambridge, 1997.

[3] Adleman, Leonard M.; McCurley, Kevin S.
Open problems in number-theoretic complexity. II.
Algorithmic number theory (Ithaca, NY, 1994), 291--322,
Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994. 95m:11142
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.4877

[4] sci.math.research, 1996/07/17
poly FLT, abc theorem, Wronskian formalism [was: Entire solutions of f^2+g^2=1]
http://groups.google.com/group/sci.math/msg/4a53c1e94f1705ed
http://google.com/[email protected]