Mason–Stothers theorem

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 gives an example where the maximum degree of the three polynomials ( and as the summands on the left hand side, and as the right hand side) is p, but the degree of the radical is only 2.
• Taking and 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 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

gave the following elementary proof of the Mason–Stothers theorem.{{citation|mr=1781918 |doi=10.1007/s000170050074

Step 1. The condition implies that the Wronskians , 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 then so a divides a′ (as a and b are coprime) so (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 :: 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} \le \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 .{{citation

There is a further generalization, due independently to J. F. Voloch{{citation and to W. D. Brownawell and D. W. Masser,{{citation that gives an upper bound for n-variable S-unit equations 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} \le \frac{1}{2}n(n-1)\max\bigl{|S| + 2g - 2,0\bigr}.

References

References

1. Mason, R. C.. (1984). "Diophantine Equations over Function Fields". Cambridge University Press.
2. Lang, Serge. (2002). "Algebra". Springer-Verlag.