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: \(\phi= F/G\).

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 |