Zeros of analytic functions, with or without multiplicities

Abstract

Abstract. The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation a + b = c. It provides a lower bound on the number of distinct zeros of the polynomial abc in terms of deg a, deg b and deg c. We extend this to general analytic functions living on a reasonable bounded domain Ω ⊂ C, rather than on the whole of C. The estimates obtained are sharp, for any Ω, and a generalization of the original result on polynomials can be recovered from them by a limiting argument. IntroductionGiven a polynomial p (in one complex variable), write deg p for the degree of p and N (p) = N C (p) for the number of its distinct zeros in C. The so-called abc theorem, also known as Mason's or Mason-Stothers' theorem, reads as follows.Theorem A. Suppose a, b and c are relatively prime polynomials, not all constants, satisfying a + b = c. ThenOf course, the two sides of (1.1) are positive integers, so (1.1) is equivalent to saying that the left-hand side does not exceed N (abc) − 1.More general variants of this result can be found in Mason's book [15], while the current version is essentially due to Stothers [19]. Various approaches to and consequences of Theorem A are discussed in [10,11,14,17], the most impressive corollary being probably Fermat's Last Theorem for polynomials. The argument leading from abc to Fermat is delightfully simple and elegant, so we take the liberty of reproducing it here.To prove that there are no nontrivial polynomial solutions to the Fermat equation P n + Q n = R n for n ≥ 3, apply the abc theorem with a = P n , b = Q n and c = R n . Write

Wronskians and deep zeros of holomorphic functions

“…In conclusion, we mention that this paper shares some common features with the author's recent work in [17,18,19], where Wronskians were employed in connection with function-theoretic analogues of the so-called abc conjecture. Also, a portion of our current Section 4 was previously announced in [20], in rather a sketchy form.…”

𝐴𝐵𝐶-type estimates via Garsia-type norms

“…As in [5], we begin by verifying that the ratio W B n /B is analytic on D (and in fact on D ∪ T). We need not worry about the zeros of B whose multiplicity is at most n, since these are obviously killed by the numerator, W B n .…”

“…Various approaches to Theorem A can be found in [7,8,10,12]. Let us also mention the following generalization involving any finite number of polynomials; see [1,5,8] for this and other related results.…”

“…Also, in [4], inequality (1.6) was supplemented with a certain alternative estimate, which we do not cite here. Further developments, as contained in [5], included the situation where the functions f j are merely analytic on D and suitably smooth up to T (but not necessarily analytic on D ∪ T); in particular, the case of infinitely many zeros was dealt with. In addition, other -fairly general -domains were considered in place of the disk.…”

A Stothers–Mason theorem with a difference radical

“…The Stothers-Mason theorem is a counterpart of the abc conjecture in number theory, while its consequence described above is Fermat's last theorem for polynomials (see, e.g., [24,25]). Fermat type functional equations, such as (1.1) and its generalizations, have been studied over many function fields [10,17,21] (see also, e.g., [15] and the references therein). For instance, if…”