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: 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, the identity gives an example where the maximum degree of the three polynomials 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 so . 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 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 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
