Démonstrations géométriques de lemmes de zéros. I, II. (Geometric proofs of zero estimates. I, II).

*(French)*Zbl 0578.10040
Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1981/82, Prog. Math. 38, 201-205 (1983); Approximations diophantiennes et nombres transcendants, Colloq. Luminy/Fr. 1982, Prog. Math. 31, 191-197 (1983).

[For the entire collections see Zbl 0514.00012 and Zbl 0504.00005. The same review was published in Mathematical Reviews.]

Let G be a commutative group variety embedded in projective space, and let \(\Gamma\) be a finitely generated subgroup of G. In the first of these two notes the author gives a very fast proof of the main result of G. Wüstholz and the reviewer [Invent. Math. 64, 489-516 (1981; Zbl 0467.10025)] about homogeneous polynomials vanishing on large parts of \(\Gamma\). Although the proof is essentially a translation of the algebraic version in the paper of Wüstholz and the reviewer into geometric terms, it enables many of the technicalities to be avoided, and furthermore the constants are slightly improved.

In the second note the result is extended to arbitrary subsets. Namely there is a constant c, depending only on the embedding of G, with the following property. Suppose \(n=\dim G\geq 1\), and for a subset \(\Sigma\) of G containing the origin let \(\Sigma^{(n)}\) denote the set of elements \(\sigma_ 1+...+\sigma_ n\) for \(\sigma_ 1,...,\sigma_ n\) in \(\Sigma\). Let \(D\geq 1\), and suppose that for any algebraic subgroup H of G of dimension \(r<n\) there are more than \(cD^{n-r}\) elements of \(\Sigma\) mutually incongruent modulo H. Then a homogeneous polynomial of degree at most D that vanishes on \(\Sigma^{(n)}\) must necessarily vanish on G. The author calculates c when the embedding is such that the group law \(G\times G\to G\) extends to a morphism \(G\times \bar G\to \bar G\) involving the Zariski closure \(\bar G.\) He also sketches the proof of a generalization to multiprojective varieties. Again the proofs are geometric.

Finally let us point out that the reviewer and G. Wüstholz have given very general results of this sort [ibid. 80, 233-267 (1985; Zbl 0564.10041)]. However, these results also involve one-parameter multiplicities, for which the algebraic proofs still seem necessary.

Let G be a commutative group variety embedded in projective space, and let \(\Gamma\) be a finitely generated subgroup of G. In the first of these two notes the author gives a very fast proof of the main result of G. Wüstholz and the reviewer [Invent. Math. 64, 489-516 (1981; Zbl 0467.10025)] about homogeneous polynomials vanishing on large parts of \(\Gamma\). Although the proof is essentially a translation of the algebraic version in the paper of Wüstholz and the reviewer into geometric terms, it enables many of the technicalities to be avoided, and furthermore the constants are slightly improved.

In the second note the result is extended to arbitrary subsets. Namely there is a constant c, depending only on the embedding of G, with the following property. Suppose \(n=\dim G\geq 1\), and for a subset \(\Sigma\) of G containing the origin let \(\Sigma^{(n)}\) denote the set of elements \(\sigma_ 1+...+\sigma_ n\) for \(\sigma_ 1,...,\sigma_ n\) in \(\Sigma\). Let \(D\geq 1\), and suppose that for any algebraic subgroup H of G of dimension \(r<n\) there are more than \(cD^{n-r}\) elements of \(\Sigma\) mutually incongruent modulo H. Then a homogeneous polynomial of degree at most D that vanishes on \(\Sigma^{(n)}\) must necessarily vanish on G. The author calculates c when the embedding is such that the group law \(G\times G\to G\) extends to a morphism \(G\times \bar G\to \bar G\) involving the Zariski closure \(\bar G.\) He also sketches the proof of a generalization to multiprojective varieties. Again the proofs are geometric.

Finally let us point out that the reviewer and G. Wüstholz have given very general results of this sort [ibid. 80, 233-267 (1985; Zbl 0564.10041)]. However, these results also involve one-parameter multiplicities, for which the algebraic proofs still seem necessary.

Reviewer: D.W.Masser

##### MSC:

11J81 | Transcendence (general theory) |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |