Mason–Stothers theorem

The Mason–Stothers theorem, or simply Mason's theorem, is a mathematical theorem about polynomials, analogous to the abc conjecture for integers. It is named after W. Wilson Stothers, who published it in 1981,^[1] and R. C. Mason, who rediscovered it shortly thereafter.^[2]

The theorem states:

Let

a (t)

b (t)

, and

c (t)

be relatively prime polynomials over a field such that

a + b = c

and such that not all of them have vanishing derivative. Then

\max\{\deg(a),\deg(b),\deg(c)\} \le \deg(\operatorname{rad}(abc))-1.

Here $rad(f)$ is the product of the distinct irreducible factors of f. For algebraically closed fields it is the polynomial of minimum degree that has the same roots as $f$ ; in this case $deg(rad(f))$ gives the number of distinct roots of $f$ .^[3]

Examples

Over fields of characteristic 0 the condition that a, b, c do not all have vanishing derivative is equivalent to the condition that they are not all constant. Over fields of characteristic $p > 0$ it is not enough to assume that they are not all constant. For example, the identity $t p + 1 = (t + 1) p$ gives an example where the maximum degree of the three polynomials ( $a$ and $b$ as the summands on the left hand side, and $c$ as the right hand side) is $p$ , but the degree of the radical is only $2$ .

Taking a(t)=tⁿ and c(t) = (t+1)ⁿ gives an example where equality holds in the Mason–Stothers theorem, showing that the inequality is in some sense the best possible.

A corollary of the Mason–Stothers theorem is the analog of Fermat's last theorem for function fields: if a(t)ⁿ + b(t)ⁿ = c(t)ⁿ for a, b, c polynomials over a field of characteristic not dividing n and n>2 then either at least one of a, b, c is 0 or they are all constant.

Proof

Snyder (2000) gave the following elementary proof of the Mason–Stothers theorem.

Step 1. The condition a+b+c=0 implies that the Wronskians W(a,b) = ab′ − a′b, W(b,c), W(c,a) are all equal. Write W for their common value.

Step 2. The condition that at least 1 of the derivatives a′, b′, c′ is nonzero and a, b, c are coprime is used to show that W is nonzero. For example, if W = 0 then ab′ = a′b so a divides a′ (as a and b are coprime) so a′=0 (as deg a>deg a′ unless a is constant).

Step 3. W is divisible by each of the greatest common divisors (a, a′), (b, b′), and (c, c′). Since these are coprime it is divisible by their product, and since W is nonzero we get

deg (a, a′) + deg (b, b′) + deg (c, c′) ≤ deg W

Step 4. Substituting in the inequalities

deg (a, a′) ≥ deg a − (number of distinct roots of a)

deg (b, b′) ≥ deg b − (number of distinct roots of b)

deg (c, c′) ≥ deg c − (number of distinct roots of c)

(where the roots are taken in some algebraic closure) and

deg W ≤ deg a + deg b − 1

we find that

deg c ≤ (number of distinct roots of abc) − 1

which is what we needed to prove.

References

↑ Stothers, W. W. (1981), "Polynomial identities and hauptmoduln", Quarterly J. Math. Oxford, 2 32: 349–370, doi:10.1093/qmath/32.3.349 .
↑ Mason, R. C. (1984), Diophantine Equations over Function Fields, London Mathematical Society Lecture Note Series 96, Cambridge, England: Cambridge University Press .
↑ Lang, Serge (2002). Algebra. New York, Berlin, Heidelberg: Springer-Verlag. p. 194. ISBN 0-387-95385-X.

Snyder, Noah (2000), "An alternate proof of Mason's theorem" (PDF), Elemente der Mathematik 55 (3): 93–94, doi:10.1007/s000170050074, MR 1781918

External links

Weisstein, Eric W., "Mason's Theorem", MathWorld.
Mason-Stothers Theorem and the ABC Conjecture, Vishal Lama. A cleaned-up version of the proof from Lang's book.

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