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, and R. C. Mason, who rediscovered it shortly thereafter. The theorem states:
Let , , and  be relatively prime polynomials over a field such that  and such that not all of them have vanishing derivative. Then

Here  is the product of the distinct irreducible factors of . For algebraically closed fields it is the polynomial of minimum degree that has the same roots as ; in this case  gives the number of distinct roots of . Examples

Over fields of characteristic 0 the condition that , , and  do not all have vanishing derivative is equivalent to the condition that they are not all constant. Over fields of characteristic  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 , but the degree of the radical is only . 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 , ,  relatively prime polynomials over a field of characteristic not dividing   and  then either at least one of , , or  is 0 or they are all constant. Proof
 gave the following elementary proof of the Mason–Stothers theorem. Step 1. The condition  implies that the Wronskians , , and  are all equal. Write  for their common value. Step 2. The condition that at least one of the derivatives , , or  is nonzero and that , , and  are coprime is used to show that  is nonzero. For example, if  then  so  divides  (as  and  are coprime) so  (as  unless  is constant). Step 3.  is divisible by each of the greatest common divisors , , and . Since these are coprime it is divisible by their product, and since  is nonzero we get

Step 4. Substituting in the inequalities
 − (number of distinct roots of )
 − (number of distinct roots of )
 − (number of distinct roots of )
(where the roots are taken in some algebraic closure) and

we find that

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  be an algebraically closed field of characteristic 0, let  be a smooth projective curve
of genus , let
 be rational functions on  satisfying ,
and let
 be a set of points in  containing all of the zeros and poles of  and . Then

Here the degree of a function in  is the degree of
the map it induces from  to P1. This was proved by Mason, with an alternative short proof published the same year by J. H. Silverman
. There is a further generalization, due independently to J. F. Voloch
and to
W. D. Brownawell and D. W. Masser,
that gives an upper bound for  -variable -unit
equations  provided that
no subset of the  are -linearly dependent. Under this assumption, they prove that

References

External links

Mason-Stothers Theorem and the ABC Conjecture, Vishal Lama. A cleaned-up version of the proof from Lang's book. Theorems about polynomials