## On Davenport’s bound for the degree of $$f^ 3 - g^ 2$$ and Riemann’s existence theorem.(English)Zbl 0840.11015

Acta Arith. 71, No. 2, 107-137 (1995); addendum ibid. 74, No. 4, 387 (1996).
A theorem of H. Davenport [Norske Vid. Selsk. Forhdl. 38, 86-87 (1965; Zbl 0136.25204)] asserts that if $$f$$ and $$g$$ are non-constant polynomials in $$\mathbb{C}[T]$$ with $$f^3\neq g^2$$, then $$\deg(f^3- g^2)\geq {1\over 2} f+ 1$$. This paper shows that, for all positive integers $$n$$, there exist polynomials $$f$$ and $$g$$ of degree $$2n$$ and $$3n$$, respectively, such that $$\deg(f^3- g^2)$$ attains this lower bound. In addition, for each $$n> 0$$ the number of “essentially distinct” families of such polynomials $$f$$ and $$g$$ is computed via a $$1-1$$ correspondence with a certain set of weighted trees.
After the paper appeared, however, the author has become aware that these results were already proved by W. W. Stothers [Q. J. Math., Oxf. II. Ser. 32, 349-370 (1981; Zbl 0466.12011)].
More generally, let $$F$$ and $$G$$ be polynomials of degree $$n> 0$$ such that $$F$$ has $$h$$ (distinct) roots with multiplicities $$\mu_1,\dots, \mu_h$$ and $$G$$ has roots with multiplicities $$\nu_1,\dots, \nu_j$$. Let $$\delta= \text{gcd}(\mu_1,\dots, \mu_h, \nu_1,\dots, \nu_j)$$. Using a theorem of W. W. Stothers (op. cit.) (this theorem is more widely attributed to R. C. Mason: it is a predecessor to the $$abc$$ conjecture), the author shows that $\deg(F- G)\geq \max(n- n/\delta, n- h- k+ 1).$ Let $$n$$ be a positive integer, and let $$\mu_1,\dots, \mu_h$$ and $$\nu_1,\dots, \nu_j$$ be collections of positive integers with $$\mu_1+\cdots+\mu_h= \nu_1+\cdots \nu_j= n$$. The paper then shows that there exist polynomials $$F$$ and $$G$$ of degree $$n$$, having $$\mu_i$$ and $$\nu_i$$ as their respective sequences of multiplicities of roots, such that $$\deg(F- G)$$ is equal to the above lower bound. This is proved using the Riemann existence theorem to construct a map $$\phi: \mathbb{P}^1\to \mathbb{P}^1$$ of degree $$n$$ with suitable properties. Then $$F$$ and $$G$$ can be taken to be homogeneous coordinates of this map: $$\phi= F/G$$.
Reviewer: P.Vojta (Berkeley)

### MSC:

 11D75 Diophantine inequalities 11R58 Arithmetic theory of algebraic function fields 14G25 Global ground fields in algebraic geometry

### Citations:

Zbl 0136.25204; Zbl 0466.12011
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