A Small Value Estimate For

Abstract

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...

A small value estimate in dimension two involving translations by rational points

“…As we will see below, this improvement allows us to gain an order of magnitude for the application that we have in view. The proof of the lemma follows that of [9,Prop. 3.3].…”

“…The proof of our main theorem is, for its first part, similar to that of [9, Theorem 1.1] and we take advantage of this by simply indicating how the corresponding arguments of [9] have to be modified. However, there are three new ingredients in the proof which have independent interest and could be applied to other situations.…”

“…This result allows us to improve the constraint on ν from [5, Theorem 1.1.5] and to obtain a best possible constraint when σ ≥ 3/2 as mentioned above. However, it does not seem to apply to the situation of [9]. It would be interesting to know if the constraint on ν in our main theorem could be improved to ν > 2 + β − σ for any value of σ in the range 1 ≤ σ < 2.…”

“…It implies the interpolation estimates that we need here as well as the one of [9]. The second ingredient is a simple idea which avoids, in the present situation, the complicated division process behind [9,Proposition 3.7], also used in [5]. It allows us to remove the condition ν > 2 + σ from [5, Theorem 1.1.5].…”

“…The first one, in Section 3, is an improvement on a formula of Mahler from [3]. It implies the interpolation estimates that we need here as well as the one of [9]. The second ingredient is a simple idea which avoids, in the present situation, the complicated division process behind [9,Proposition 3.7], also used in [5].…”