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Zbl 0596.10032
Viola, Carlo
On Dyson's lemma. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 105-135 (1985). ISSN 0391-173X

Let P(X,Y) be a non-zero polynomial with complex coefficients and of bi- degree m,n such that $(\partial\sp{i+j}/\partial X\sp i \partial Y\sp j) P(\alpha\sb k,\beta\sb k)=0$ for all (i,j) satisfying $\theta$ i/m$+\psi j/n<t\sb k$ $(k=1,...,M)$ for points $(\alpha\sb k,\beta\sb k)$ with the $\alpha\sb k$ pairwise distinct. Set $$ f(t)=f(t,\theta,\psi)=\int\sp{1}\sb{0}\int\sp{1}\sb{0}dx dy\quad (integrated\quad on\quad \theta x+\psi y\le t) $$ observing that $f(t)\sim (\theta \psi)\sp{-1}t\sp 2$. Then $$ \sum\sp{M}\sb{k=1}f(t\sb k)\quad \le \quad 1+\max (M/2-1,0)\quad \min (n/m,m/n). $$ This result, which is implied by the somewhat more general result proved in this paper, may be viewed as an analogue for polynomials in two variables of the trivial remark that a nonzero polynomial in one variable and of degree m has at most m zeros counted according to multiplicity. Its role in the theory of diophantine approximation is to yield a guarantee that a construction of a critical auxiliary number indeed yields a nonzero quantity. \par A result of the above kind appeared in a paper of {\it F. J. Dyson} [Acta Math. 79, 225-240 (1947; Zbl 0030.02101)]. Its present refinement is obtained by {\it E. Bombieri} [Acta Math. 148, 255-296 (1982; Zbl 0505.10015)] where it is applied to yield effective approximation results in a wide class of number fields. \par The present paper provides an elegant intrinsic proof of the Lemma using methods of classical algebraic geometry which make the result appear somewhat more natural than it does on first acquaintance with the earlier proofs. Since the result is slightly less restrictive in its admissibility condition on the points $(\alpha\sb k,\beta\sb k)$ than is Bombieri's version the application stated is correspondingly more general. \par The corresponding Lemma for polynomials in several variables has been obtained by {\it H. Esnault} and {\it E. Viehweg} [Invent. Math. 78, 445-490 (1984; Zbl 0532.10020)] with their proof inter alia employing sophisticated techniques of modern algebraic geometry.
[A.J.van der Poorten]

MSC 2000:: *11J68 Approximation to algebraic numbers

Keywords: rational approximations to algebraic numbers; number of zeros; Dyson lemma; effective results; polynomials in two variables

Citations: Zbl 0545.10021; Zbl 0030.02101; Zbl 0505.10015; Zbl 0532.10020

Cited in: Zbl 0774.11034 Zbl 0773.11043 Zbl 0666.10024

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