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Mason–Stothers theorem

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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 W. Wilson Stothers, who published it in 1981, and R. C. Mason, who rediscovered it shortly thereafter.

Contents

The theorem states:

Let a(t), b(t), and c(t) be relatively prime polynomials over a field such that a + b = c and such that not all of them have vanishing derivative. Then max { deg ⁡ ( a ) , deg ⁡ ( b ) , deg ⁡ ( c ) } ≤ deg ⁡ ( rad ⁡ ( a b c ) ) − 1.

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

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, the identity tp + 1 = (t + 1)p gives an example where the maximum degree of the three polynomials (a and b as the summands on the left hand side, and c as the right hand side) is p, but the degree of the radical is only 2.
  • Taking a(t) = tn and c(t) = (t+1)n 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 a(t)n + b(t)n = c(t)n 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

    Snyder (2000) gave the following elementary proof of the Mason–Stothers theorem.

    Step 1. The condition a + b + c = 0 implies that the Wronskians W(a,b) = ab′ − a′b, 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 W = 0 then ab′ = a′b so a divides a′ (as a and b are coprime) so a′ = 0 (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.

    References

    Mason–Stothers theorem Wikipedia
    (Text) CC BY-SA


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