Mason's Theorem

Let there be three Polynomials $a(x)$, $b(x)$, and $c(x)$ with no common factors such that

$$a(x)+b(x)=c(x).$$

Then the number of distinct Roots of the three Polynomials is one or more greater than their largest degree. The theorem was first proved by Stothers (1981).

Mason's theorem may be viewed as a very special case of a Wronskian estimate (Chudnovsky and Chudnovsky 1984). The corresponding Wronskian identity in the proof by Lang (1993) is

$$c^3*W(a,b,c) = W(W(a,c),W(b,c)),$$

so if $a$, $b$, and $c$ are linearly dependent, then so are $W(a,c)$ and $W(b,c)$. More powerful Wronskian estimates with applications toward diophantine approximation of solutions of linear differential equations may be found in Chudnovsky and Chudnovsky (1984) and Osgood (1985).

The rational function case of Fermat's Last Theorem follows trivially from Mason's theorem (Lang 1993, p. 195).

See also abc Conjecture

References

Chudnovsky, D. V. and Chudnovsky, G. V. ``The Wronskian Formalism for Linear Differential Equations and Padé Approximations.'' Adv. Math. 53, 28-54, 1984.

Lang, S. ``Old and New Conjectured Diophantine Inequalities.'' Bull. Amer. Math. Soc. 23, 37-75, 1990.

Lang, S. Algebra, 3rd ed. Reading, MA: Addison-Wesley, 1993.

Osgood, C. F. ``Sometimes Effective Thue-Siegel-Roth-Schmidt-Nevanlinna Bounds, or Better.'' J. Number Th. 21, 347-389, 1985.

Stothers, W. W. ``Polynomial Identities and Hauptmodulen.'' Quart. J. Math. Oxford Ser. II 32, 349-370, 1981.