## On the product theorem. (Sur le théorème du produit.)(English)Zbl 1047.14003

Faltings’ product Theorem is a generalization of Roth’s Lemma [G. Faltings, Ann. Math. (2) 133, No. 3, 549–576 (1991; Zbl 0734.14007)]. It states that if the zeros of index $$\sigma$$ of a multihomogeneous polynomial $$P$$ have a component $$Z$$ in common with the set of zeros of index $$\sigma+\epsilon$$, then $$Z$$ is a product. The most useful corollary is for $$\sigma=0$$. After the work of M. Nakamaye [Bull. Soc. Math. Fr. 123, No. 2, 155–188 (1995; Zbl 0841.11037)], P. Philippon [Bull. Aust. Math. Soc. 59, No. 2, 323–334 (1999; Zbl 0927.11040)] has shown how to deduce this corollary from his zero estimate [Rocky Mt. J. Math. 26, No. 3, 1069–1088 (1996; Zbl 0893.11027)]. Effective versions due to J.-H. Evertse [Acta Arith. 73, No. 3, 215–248 (1995; Zbl 0857.11034)] and R. Ferretti [Forum Math. 8, No. 4, 401–427 (1996; Zbl 0860.11038)] give upper bounds for the degree and height of $$Z$$.
Here the author proves a new sharp effective version of this result by relaxing the hypotheses on the degrees of $$P$$. The improvement compared with the previous estimates is due to the use of Samuel’s multiplicity, like in Philippon’s work, in place of the length, while the improvement with respect to Philippon’s result is due to the use of a multiprojective height [chap. 5 and 7 of Yu. V. Nesterenko and P. Philippon (eds.), Introduction to algebraic independence theory (2001; Zbl 0966.11032)], instead of a reduction to the projective situation.

### MSC:

 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14G05 Rational points 11J81 Transcendence (general theory) 11J13 Simultaneous homogeneous approximation, linear forms 11J20 Inhomogeneous linear forms
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### References:

 [1] Bost, J.-B., Gillet, H., Soulé, C., Heights of projective varieties and positive Green forms. J. Amer. Math. Soc.7 (1994), 903-1027. · Zbl 0973.14013 [2] Bourbaki, N., Algèbre commutative, chapitre VIII. Masson, 1983. [3] Brownawell, W.D., Note on a paper of P. Philippon. Michigan Math. J.34 (1987), 461-464. · Zbl 0634.14002 [4] Evertse, J.-H., An explicit version of Faltings’ product theorem and an improvement of Roth’s lemma. Acta Arithm.73 (1995), 215-248. · Zbl 0857.11034 [5] Faltings, G., Diophantine approximation on abelian varieties. Ann. of Math.133 (1991), 549-576. · Zbl 0734.14007 [6] Ferretti, R., An effective version of Faltings’ product theorem. Forum Math.8 (1996), 401-427. · Zbl 0860.11038 [7] Hartshorne, R., Algebraic geometry. G.T.M. 52, Springer-Verlag, 1977. · Zbl 0367.14001 [8] Nakamaye, M., Multiplicity estimates and the product theorem. Bull. Soc. Math. France123 (1995), 155-188. · Zbl 0841.11037 [9] Philippon, P., Sur des hauteurs alternatives III. J. Math. Pures Appl.74 (1995), 345-365. · Zbl 0878.11025 [10] Philippon, P., Nouveaux lemmes de zéros dans les groupes algébriques commutatifs. Rocky Mountain J. Math.26 (1996), 1069-1088. · Zbl 0893.11027 [11] Philippon, P., Quelques remarques sur des questions d’approximation diophantienne. Bull. Austral. Math. Soc.59 (1999), 332-334. · Zbl 0927.11040 [12] Rémond, G., Élimination multihomogéne (chapitre 5) et Géométrie diophantienne multi-projective (chapitre 7). Dans: Introduction to algebraic independence theory (édité par Y. Nesterenko et P. Philippon), , Springer-Verlag, 2001.
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