Mason–Stothers theorem explained

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,, and be relatively prime polynomials over a field such that and such that not all of them have vanishing derivative. Then

max \{\deg(a),\deg(b),\deg(c)\} \le \deg(\operatorname{rad}(abc)) -1 . 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 .^[3]

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.^[4]

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 curveof genus, let

a,b\in k(C) be rational functions on satisfying a + b =1 ,and let be a set of points in containing all of the zeros and poles of and .Thenmax l \{ \deg(a),\deg(b) r \} \le max l \{|S| + 2 g - 2 , 0 r \}. Here the degree of a function in is the degree ofthe 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. Volochand toW. D. Brownawell and D. W. Masser,that gives an upper bound for -variable -unitequations provided thatno subset of the are -linearly dependent. Under this assumption, they prove that

max l \{ \deg(a _{1),\ldots,\deg(a} _{n)} r \} \le (1)/(2) n(n -1 ) max l \{|S| + 2 g - 2 , 0 r \}.

External links

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

Notes and References

1. .
2. .
3. Book: Serge Lang

   . Lang, Serge . Serge Lang . Algebra. Springer-Verlag . New York, Berlin, Heidelberg . 2002 . 0-387-95385-X. 194.

4. .