Interpolation on algebraic groups.

*(English)*Zbl 1080.14054One way to argue transcendence is to construct a linear combination of evaluations of derivations which takes small algebraic values on many functions. It is then argued via interpolation that this combination is in fact zero. This approach is the so-called dual to the usual process of constructing an auxiliary function and showing that is must be zero.

It is the interpolation part of the first argument which is the focus of this paper. The case of interpolation on a commutative algebraic group with no multiplicities is a result of [D. Masser, Progr. Math. 31, 151–171 (1983; Zbl 0579.14038)].

Here the author provides an interpolation lemma, a generalization to Masser’s, for all commutative algebraic groups over \(\mathbb{C}\). A sketch of the statement of the result is as follows. Let \(G\) be a connected commutative algebraic group. Let \(D,\) \(\underline{S}:=S_{1},\dots ,S_{l},\underline{T}:=T_{1},\dots ,T_{d}\) be positive integers such that, for \(H\) a non-zero connected algebraic subgroup of \(G\) satisfying a certain bound. Then there is a homogeneous polynomial \(P\) of degree \(D\) such that \( \partial ^{\sigma }( P/X_{0}^{D}) ( \gamma ) =a_{\gamma ,\sigma }\), where \(\gamma \) is in a specific finitely generated subgroup of \(G( \mathbb{C}) \), \(\sigma \in \mathbb{N}^{d}\) with \( \sigma _{i}<T_{i}\) for all \(i\), and \(a_{\gamma ,\sigma }\in \mathbb{C}.\) The above can also be stated in terms of cohomology. For \(H\) a non-zero connected algebraic subgroup of \(G\) satisfying a certain bound (which is the same as the one above up to a positive constant) we have \(H^{1}( \bar{G} ,\mathcal{I}_{\underline{S},\underline{T},D}) =0,\) where \(\mathcal{I}_{ \underline{S},\underline{T},D} =\mathcal{J}_{\underline{S},\underline{T} }\otimes \mathcal{O}( D) \) for some certain ideal sheaf \(\mathcal{ J}_{\underline{S},\underline{T}}.\)

Furthermore, there is a statement of the main result in terms of functionals, which says that the result is equivalent to the fact that there is no non-zero functional \(\eta \) such that \(\eta ( P) =0\) for all \(P.\) Finally, after some counting arguments it is shown that for a non-zero functional \(\eta \) which vanishes on all \(P\) there exists a non-zero connected algebraic subgroup \(H\) of \(G\) which satisfies another boundedness condition. The lower bound given is shown to be the best possible.

It is the interpolation part of the first argument which is the focus of this paper. The case of interpolation on a commutative algebraic group with no multiplicities is a result of [D. Masser, Progr. Math. 31, 151–171 (1983; Zbl 0579.14038)].

Here the author provides an interpolation lemma, a generalization to Masser’s, for all commutative algebraic groups over \(\mathbb{C}\). A sketch of the statement of the result is as follows. Let \(G\) be a connected commutative algebraic group. Let \(D,\) \(\underline{S}:=S_{1},\dots ,S_{l},\underline{T}:=T_{1},\dots ,T_{d}\) be positive integers such that, for \(H\) a non-zero connected algebraic subgroup of \(G\) satisfying a certain bound. Then there is a homogeneous polynomial \(P\) of degree \(D\) such that \( \partial ^{\sigma }( P/X_{0}^{D}) ( \gamma ) =a_{\gamma ,\sigma }\), where \(\gamma \) is in a specific finitely generated subgroup of \(G( \mathbb{C}) \), \(\sigma \in \mathbb{N}^{d}\) with \( \sigma _{i}<T_{i}\) for all \(i\), and \(a_{\gamma ,\sigma }\in \mathbb{C}.\) The above can also be stated in terms of cohomology. For \(H\) a non-zero connected algebraic subgroup of \(G\) satisfying a certain bound (which is the same as the one above up to a positive constant) we have \(H^{1}( \bar{G} ,\mathcal{I}_{\underline{S},\underline{T},D}) =0,\) where \(\mathcal{I}_{ \underline{S},\underline{T},D} =\mathcal{J}_{\underline{S},\underline{T} }\otimes \mathcal{O}( D) \) for some certain ideal sheaf \(\mathcal{ J}_{\underline{S},\underline{T}}.\)

Furthermore, there is a statement of the main result in terms of functionals, which says that the result is equivalent to the fact that there is no non-zero functional \(\eta \) such that \(\eta ( P) =0\) for all \(P.\) Finally, after some counting arguments it is shown that for a non-zero functional \(\eta \) which vanishes on all \(P\) there exists a non-zero connected algebraic subgroup \(H\) of \(G\) which satisfies another boundedness condition. The lower bound given is shown to be the best possible.

Reviewer: Alan Koch (Decatur)