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 Walter 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

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

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, and 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, considered as polynomials over some field of characteristic p, 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 relatively prime polynomials over a field of characteristic not dividing n and n > 2 then either at least one of a, b, or c is 0 or they are all constant.

Proof

Snyder (2000) gave the following elementary proof of the Mason–Stothers theorem.^[4]

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

Step 2. The condition that at least one of the derivatives a′, b′, or c′ is nonzero and that a, b, and 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.

Generalizations

There is a natural generalization in which the ring of polynomials is replaced by a one-dimensional function field. Let k be an algebraically closed field of characteristic 0, let C/k be a smooth projective curve of genus g, let

${\displaystyle a,b\in k(C)}$  be rational functions on C satisfying ${\displaystyle a+b=1}$ ,

and let S be a set of points in C(k) containing all of the zeros and poles of a and b. Then

${\displaystyle \max {\bigl \{}\deg(a),\deg(b){\bigr \}}\leq \max {\bigl \{}|S|+2g-2,0{\bigr \}}.}$ 

Here the degree of a function in k(C) is the degree of the map it induces from C to P¹. This was proved by Mason, with an alternative short proof published the same year by J. H. Silverman .^[5]

There is a further generalization, due independently to J. F. Voloch^[6] and to W. D. Brownawell and D. W. Masser,^[7] that gives an upper bound for n-variable S-unit equations a₁ + a₂ + ... + a_n = 1 provided that no subset of the a_i are k-linearly dependent. Under this assumption, they prove that

${\displaystyle \max {\bigl \{}\deg(a_{1}),\ldots ,\deg(a_{n}){\bigr \}}\leq {\frac {1}{2}}n(n-1)\max {\bigl \{}|S|+2g-2,0{\bigr \}}.}$ 

References

1. Stothers, W. W. (1981), "Polynomial identities and hauptmoduln", Quarterly J. Math. Oxford, 2, 32: 349–370, doi:10.1093/qmath/32.3.349.
2. Mason, R. C. (1984), Diophantine Equations over Function Fields, London Mathematical Society Lecture Note Series, vol. 96, Cambridge, England: Cambridge University Press.
3. Lang, Serge (2002). Algebra. New York, Berlin, Heidelberg: Springer-Verlag. p. 194. ISBN 0-387-95385-X.
4. Snyder, Noah (2000), "An alternate proof of Mason's theorem" (PDF), Elemente der Mathematik, 55 (3): 93–94, doi:10.1007/s000170050074, MR 1781918.
5. Silverman, J. H. (1984), "The S-unit equation over function fields", Proc. Camb. Philos. Soc., 95: 3–4
6. Voloch, J. F. (1985), "Diagonal equations over function fields", Bol. Soc. Bras. Mat., 16: 29–39
7. Brownawell, W. D.; Masser, D. W. (1986), "Vanishing sums in function fields", Math. Proc. Cambridge Philos. Soc., 100: 427–434

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.