The primary reason that function field arithmetic is simpler than number field arithmetic is due to the existence of nontrivial derivations. With the availability of derivatives many things simplify.
E.g. for polynomials derivatives yield easy algorithms for squarefree testing, squarefree part, etc. Contrast this to the integer case. 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.
From derivatives also come Wronskians and associated measures of independence (excerpted below). For example, this is what is at the heart of Mason's trivial high-school level proof of the ABC theorem for polynomials - which is a difficult important open problem for numbers. From Mason's theorem follows immediately a trivial two-line proof of FLT for polynomials. If there existed some sort of analogous "derivative for integers" that yielded the corresponding ABC theorem, then it would yield an analogous trivial proof of FLT for integers (more precisely it would yield asymptotic FLT, i.e. FLT for all sufficiently large exponents).
Such observations have motivated searches for "arithmetic analogues of derivations". For example, see Buium's paper by that name in Jnl. Algebra, 198, 1997, 290-99, and see his book Arithmetic differential equations.
poly FLT, abc theorem, Wronskian formalism [was: Entire solutions of f^2+g^2=1]
Posted: Jul 17, 1996 12:13 AM
Click to see the message monospaced in plain text Plain Text Click to reply to this topic Reply
"Harold P. Boas" wrote to sci.math.research on 7/3/96:
:Robert Israel wrote:
:> Alan Horwitz writes:
:> |> I am interested in all entire solutions f and g to f^2+g^2=1.
:> |> I remember seeing this somewhere, but I cannot recall where.
:> I've also seen this before, in fact I recall assigning it as homework
:> to one of my classes, but I don't recall the source. The solutions are ...
:Robert B. Burckel gives some history about this problem in his
:comprehensive book An Introduction to Classical Complex Analysis,
:volume 1 (Academic Press, 1979). In Theorem 12.20, pages 433-435,
:he shows that the equation f^n+g^n=1 has no nonconstant entire
:solutions when the integer n exceeds 2; when n=2, the solution
:is as given by R. Israel in his post. ... (papers of Fred Gross)
Note that the rational function case of FLT follows trivially from
Mason's abc theorem, e.g. see Lang's Algebra, 3rd Ed. p. 195 for a
short elementary (high-school level) proof of both. Chebyshev also
gave a proof of FLT for poly's via the theory of integration in
finite terms, e.g. see p. 145 of Shanks' "Solved and Unsolved Problems
in Number Theory", or Ritt's "Integration in Finite Terms", p. 37.
The Chebyshev result is actually employed as a subroutine of Macsyma's
integration algorithm (implemented decades ago by Joel Moses). Via abc
a related result of Dwork is also easily proved: if A,B,C are fixed
poly's then all coprime poly solutions of AX^a+BY^b+C*Z^c = 0
have bounded degrees provided 1/a+1/b+1/c < 1. Other applications
in both number and function fields may be found in Lang's survey .
Mason's abc theorem may be viewed as a very special instance of a
Wronskian estimate: in Lang's proof the corresponding Wronskian
identity is c^3*W(a,b,c) = W(W(a,c),W(b,c)), thus if a,b,c are
linearly dependent then so are W(a,c),W(b,c); the sought bounds
follow upon multiplying the latter dependence relation through by
N0 = r(a)*r(b)*r(c), where r(x) = x/gcd(x,x').
More powerful Wronskian estimates with applications toward
diophantine approximation of solutions of LDEs may be found in
the work of the Chudnovsky's 1 and C. Osgood 2. References
to recent work may be found (as usual) by following MR citations
to these papers in the MathSci database.
I have not seen mention of this Wronskian view of Mason's abc theorem.
Although elementary, it deserves attention since it connects the abc
theorem with the general unified viewpoint of the Wronskian formalism
as proposed by the Chudnovsky's and others.
1 Chudnovsky, D. V.; Chudnovsky, G. V.
The Wronskian formalism for linear differential equations and Pade
approximations. Adv. in Math. 53 (1984), no. 1, 28--54.
86i:11038 11J91 11J99 34A30 41A21
2 Osgood, Charles F.
Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better.
J. Number Theory 21 (1985), no. 3, 347--389. 87f:11046 11J61 12H05
 Lang, Serge
Old and new conjectured Diophantine inequalities. Bull. Amer. Math. Soc.
(N.S.) 23 (1990), no. 1, 37--75. 90k:11032 11D75 11-02 11D72 11J25