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Connecting interpolation and multiplicity estimates in commutative algebraic groups. (English) Zbl 1347.11054
Let \(G\) be a commutative algebraic group and \(\Gamma\) a finitely generated subgroup of \(G\). In transcendental number theory, an important tool is a zero estimate (or multiplicity estimate), which produces a lower bound for the degree of a nonzero polynomial vanishing (with multiplicity) on some finite subset of \(\Gamma\) [D. W. Masser and G. Wüstholz, Invent. Math. 64, 489–516 (1981; Zbl 0467.10025); P. Philippon, Bull. Soc. Math. Fr. 114, 355–383 (1986; Zbl 0617.14001); ibid. 115, 397–398 (1987; Zbl 0634.14001); Rocky Mt. J. Math. 26, No. 3, 1069–1088 (1996; Zbl 0893.11027)].
Another useful tool is an interpolation estimate which produces an upper bound for a polynomial taking given values at the points of such a finite subset. [D. Roy, J. Number Theory 94, No. 2, 248–285 (2002; Zbl 1010.11039); S. Fischler, Compos. Math. 141, No. 4, 907–925 (2005; Zbl 1080.14054); M. Nakamaye, in: Number theory, analysis and geometry. In memory of Serge Lang. Berlin: Springer. 475–498 (2012; Zbl 1268.11097); S. Fischler and M. Nakamaye, Ann. Inst. Fourier 64, No. 3, 1269–1289 (2014; Zbl 1330.14076)].
Multiplicity estimates are used together with auxiliary functions, interpolation estimates with auxiliary functionals.
Each of these two auxiliary results involves some algebraic subgroup of \(G\) which is responsible for a possible obstruction. In the present paper, the authors introduce a chain of algebraic subgroups of \(G\) which control the distribution of these points. This gives a unified treatment of the two auxiliary results and provides a bridge between multiplicity estimates and interpolation lemmas.

11J81 Transcendence (general theory)
14L10 Group varieties
11J95 Results involving abelian varieties
14L40 Other algebraic groups (geometric aspects)
Full Text: DOI
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