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

\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 $a (t) = t n$ 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') \leq deg W .

Step 4. Substituting in the inequalities

deg (a, a') \geq deg a

− (number of distinct roots of

a

)

deg (b, b') \geq deg b

− (number of distinct roots of

b

)

deg (c, c') \geq deg c

− (number of distinct roots of

c

)

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

deg W \leq deg a + deg b - 1

we find that

deg c \leq (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

a,b\in k(C)

be rational functions on

C

satisfying

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

\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 $a 1 + a 2 + ... + a n = 1$ provided that no subset of the $a i$ are $k$ -linearly dependent. Under this assumption, they prove that

\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

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, vol. 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.
Silverman, J. H. (1984), "The S-unit equation over function fields", Proc. Camb. Philos. Soc., 95: 3–4
Voloch, J. F. (1985), "Diagonal equations over function fields", Bol. Soc. Bras. Mat., 16: 29–39
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.

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[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