Mason–Stothers theorem

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

$\max \{\mathrm{deg}(a),\mathrm{deg}(b),\mathrm{deg}(c)\}\le \mathrm{deg}(\mathrm{rad}(abc))-1.$

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

Examples

• Over fields of characteristic 0 the condition that Template:Mvar, Template:Mvar, and Template:Mvar do not all have vanishing derivative is equivalent to the condition that they are not all constant. Over fields of characteristic Template:Math 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 Template:Math gives an example where the maximum degree of the three polynomials (Template:Mvar and Template:Mvar as the summands on the left hand side, and Template:Mvar as the right hand side) is Template:Mvar, but the degree of the radical is only Template:Math.
• Taking Template:Math and Template:Math 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 Template:Math for Template:Mvar, Template:Mvar, Template:Mvar relatively prime polynomials over a field of characteristic not dividing Template:Mvar and Template:Math then either at least one of Template:Mvar, Template:Mvar, or Template:Mvar is 0 or they are all constant.

Proof

Template:Harvtxt gave the following elementary proof of the Mason–Stothers theorem.

Step 1. The condition Template:Math implies that the Wronskians Template:Math, Template:Math, and Template:Math are all equal. Write Template:Mvar for their common value.

Step 2. The condition that at least one of the derivatives Template:Math, Template:Math, or Template:Math is nonzero and that Template:Mvar, Template:Mvar, and Template:Mvar are coprime is used to show that Template:Mvar is nonzero. For example, if Template:Math then Template:Math so Template:Mvar divides Template:Math (as Template:Mvar and Template:Mvar are coprime) so Template:Math (as Template:Math unless Template:Mvar is constant).

Step 3. Template:Mvar is divisible by each of the greatest common divisors Template:Math, Template:Math, and Template:Math. Since these are coprime it is divisible by their product, and since Template:Mvar is nonzero we get

Template:Math

Step 4. Substituting in the inequalities

Template:Math − (number of distinct roots of Template:Mvar)

Template:Math − (number of distinct roots of Template:Mvar)

Template:Math − (number of distinct roots of Template:Mvar)

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

Template:Math

we find that

Template:Math

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 Template:Mvar be an algebraically closed field of characteristic 0, let Template:Math be a smooth projective curve of genus Template:Mvar, let

$a,b\in k(C)$ be rational functions on Template:Mvar satisfying $a+b=1$,

and let Template:Mvar be a set of points in Template:Math containing all of the zeros and poles of Template:Mvar and Template:Mvar. Then

$\max \{\mathrm{deg}(a),\mathrm{deg}(b)\}\le \max \{|S|+2g-2,0\}.$

Here the degree of a function in Template:Math is the degree of the map it induces from Template:Mvar 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 Template:Mvar-variable Template:Mvar-unit equations Template:Math provided that no subset of the Template:Math are Template:Mvar-linearly dependent. Under this assumption, they prove that

$\max \{\mathrm{deg}(a_1),\dots ,\mathrm{deg}(a_n)\}\le \frac{1}{2}n(n-1)\max \{|S|+2g-2,0\}.$

External links

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