The abc Theorem of Polynomials

Let be an algebraically closed field of characteristic . Instead of studying the polynomial ring as a whole, we pay a little more attention to each polynomial. A reasonable thing to do is to count the number of distinct zeros. We define For example, If , we have . It seems we are diving into calculus but actually there is still a lot of algebra.

The abc of Polynomials

Theorem 1 (Mason-Stothers). Let be polynomials such that and . Then

Proof. Putting and , we have This implies

We interrupt the proof here for some good reasons. Rational functions of the form remind us of the chain rule applied to . In the context of calculus, we have . On the ring , we define to be the formal derivative morphism. Then this endomorphism extends to by On (read: the multiplicative group of the rational function field ), we define the logarithm derivative It follows that Also observe that, just as in calculus, if is a constant function, then . Now we write Then it follows that Now we can be back to the proof.

Proof (continued). Since is algebraically closed, We see, for example Therefore Likewise Combining both, we obtain Next, multiplying and by which has degree (since , these three polynomials share no root). Both and are polynomials of degrees at most (this is because for non-constant , while and are non-constant (why?); we assume for this reason).

Next we observe the degrees of and . Since , we actually have . Therefore . From the relation and the assumption that , one can find polynomial such that Taking the degrees of both sides, we see This proves the theorem.

Applications

We present some applications of this theorem.

Corollary 1 (Fermat's theorem for polynomials). Let and be relatively prime polynomials in such that not all of them are constant, and such that Then .

Alternatively one can argue the curve on .

Proof. Since and are relatively prime, we also have , and to be relatively prime. By Mason-Stothers theorem, Replacing by and , we see It follows that In this case .

Corollary 2 (Davenport's inequality). Let be non-constant polynomials such that . Then

One may discuss cases separately on whether and are coprime, and try to apply Mason-Stothers theorem respectively, and many documents only record the proof of coprime case, which is a shame. The case when and are not coprime can be a nightmare. Instead, for sake of accessibility, we offer the elegant proof given by Stothers, starting with a lemma about the degree of the difference of two polynomials.

Lemma 1. Suppose are two distinct non-constant polynomials, then

Proof. Let be the leading coefficient of a polynomial . If or , then because and .

Next suppose and . If , then by Mason-Stothers, Otherwise, suppose . Then and are coprime. Again by Mason-Stothers, Therefore On the other hand, Combining all these inequalities, we obtain what we want.

Proof (of corollary 2). Put and . If , then because . Next we assume that , or in other word, and . By lemma 1, we can write This proves the inequality.

One may also generalise the case to . But we put down some more important remarks. First of all, Mason-Stothers is originally a generalisation of Davenport's inequality (by Stothers). I personally do not think any mortal can find the original paper of Davenport's inequality, but on [Shioda 04] there is a reproduced proof using linear algebra (lemma 3.1).

For more geometrical interpretation, one may be interested in [Zannier 95], where Riemann's existence theorem is also discussed.

In Stothers's paper [Stothers 81], the author discussed the condition where the equality holds. If you look carefully you will realise his theorem 1.1 is exactly the Mason-Stothers theorem.

References / Further Reading

• [Davenport 65] H. Davenport, On , 1965. (can someone find a digital copy of this paper?)
• [Ma 84] R. C. Mason, Diophantine Equations over Function Fields, 1984.
• [Shioda 04] Tetsuji Shioda, The abc-theorem, Davenport’s inequality and elliptic surfaces, 2004 (https://www2.rikkyo.ac.jp/web/shioda/papers/esdstadd.pdf)
• [Stothers 81] W. W. Stothers, POLYNOMIAL IDENTITIES AND HAUPTMODULN, 1981. (https://doi.org/10.1093/qmath/32.3.349)
• [Zannier 95] Umberto Zannier (Venezia), On Davenport’s bound for the degree of and Riemann’s Existence Theorem, 1995. (https://eudml.org/doc/206763)