Small value estimates for the multiplicative group

Roy, Damien

doi:10.4064/aa135-4-5

Cited by 2 publications

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“…In continuation with [13] and [14], the aim of the present paper is to develop new tools for algebraic independence in situations where the traditional combination of a criterion for algebraic independence and of a zero estimate does not apply. The small value estimates that we are looking for, aim at extracting as much information as possible from the global data of a sequence of auxiliary polynomials taking many small values at points of a finitely generated subgroup of a commutative algebraic group.…”

Section: Introductionmentioning

confidence: 99%

“…An ultimate goal would be to prove the conjectural small value estimates proposed in [11] and [12] and shown there to be equivalent respectively to the standard conjecture of Schanuel and its elliptic analog. In [14] and [13], we established some small value estimates respectively for the additive group C = G a (C) and the multiplicative group C * = G m (C). The present paper deals with the group…”

Section: Introductionmentioning

confidence: 99%

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A Small Value Estimate For

Roy

2013

Mathematika

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A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product G a × G m whose underlying group of complex points is C × C * . It shows that if a certain sequence of nonzero polynomials in Z[X 1 , X 2 ] take small values at a point (ξ, η) together with their first derivatives with respect to the invariant derivation ∂/∂X 1 + X 2 (∂/∂X 2 ), then both ξ and η are algebraic over Q. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon's criterion for algebraic independence and compares favorably with constructions coming from Dirichlet's box principle.On the other hand, compared to the lower bound ν > 2 + β required by a direct application of Philippon's criterion [8, Theorem 2.11], our condition on ν represents a gain of at least τ − 1/4. Although there is room for possibly improving the conditions on β and ν in Theorem 1.1, the restriction τ ≥ 1 is crucial in order to be able to conclude that ξ and η are algebraic over Q or even that they are algebraically dependent over Q. This follows from a construction of Khintchine adapted by Philippon [8, Appendix] which shows that, for any sequence of positive real numbers (ψ D ) D≥1 , there exist algebraically independent numbers ξ, η ∈ C and a sequence of non-zero linear forms (L D ) D≥1 in Z + ZX 1 + ZX 2 satisfying L D ≤ D and |L D (ξ, η)| ≤ ψ D for each D ≥ 1. Here, we apply this result with ψ D = exp(−2D ν−τ ) assuming simply that 0 ≤ τ < 1, β > τ and ν > τ . For the corresponding point (ξ, η) ∈ G and the corresponding sequence of linear forms (L D ) D≥1 , we set P D = L ⌊4D τ ⌋ D for each D ≥ 1. Then, for each sufficiently large D, the polynomial P D ∈ Z[X 1 , X 2 ] is non-zero, has degree ≤ D, norm ≤ exp(D β ) and satisfies the condition (2). However, both ξ and η are transcendental over Q.The proof of our main result combines techniques for multiplicity and zero estimates introduced by W. D. Brownawell and D. W. Masser in [1] and by D. W. Masser in [5], together with techniques of elimination theory developed by Yu. V. Nesterenko [6,7] and P. Philippon [8], and formalized in [3] in joint work with M. Laurent. We give below a short outline of that proof. The complete argument occupies Section 7.Arguing by contradiction, we first replace each polynomial P D by an appropriate homogeneous polynomialP D of degree D, and replace the differential operator D 1 by a corresponding homogeneous operator D on C[X] := C[X 0 , X 1 , X 2 ]. For each integer D, we also define a convex body C D of the homogeneous part C[X] D of C[X] of degree D. It consists of all polynomials of C[X] D w...

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Section: Introductionmentioning

confidence: 99%