Small Value Estimates for the Additive Group

Abstract

We generalize Gel'fond's criterion for algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of ℂ, instead of just one point (one extension deals with a subgroup of ℂ×).

Small value estimates for the multiplicative group

“…If n is sufficiently large as a function of β, δ and µ, we have deg(Q) ≤ n 1−µ+δ and H(Q) ≤ exp(n β−2µ+δ ). This result is the multiplicative analog of Theorem 1.2 of [7]. To achieve such non-trivial estimates on the degree and height of Q, the requirement that P has no root in C × tor ∪ {0} is necessary.…”

“…A good illustration of the need for refined criteria, and our main motivation for this quest, is a conjectural small value estimate for the algebraic group G a × G m which is proposed in [6] and shown to be equivalent to Schanuel's conjecture. In a preceding paper [7], we explored the case of the additive group G a . Here, we turn to the multiplicative group G m .…”

“…The proof of the above result is involved but the main underlying idea is simple and is inspired by techniques from zero estimates. If a polynomial P ∈ Z[T ] takes small values at all points of the form ξ a with ξ in a subset E of C × and a in a subset A of N * , then the polynomials P (T a ) with a ∈ A take small values at all points of E. Applying Corollary 3.2 of [7], one deduces that the product ξ∈E |Q(ξ)| is small, where Q(T ) denotes the greatest common divisor in Z[T ] of the polynomials P (T a ) with a ∈ A. However, for this to be useful, we also need good upper bounds for the degree and height of Q(T ).…”

A Small Value Estimate For

“…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.…”

“…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…”