# Intersecting a plane with algebraic subgroups of multiplicative groups

Enrico Bombieri; David Masser; Umberto Zannier

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

- Volume: 7, Issue: 1, page 51-80
- ISSN: 0391-173X

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topBombieri, Enrico, Masser, David, and Zannier, Umberto. "Intersecting a plane with algebraic subgroups of multiplicative groups." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.1 (2008): 51-80. <http://eudml.org/doc/272261>.

@article{Bombieri2008,

abstract = {Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed elsewhere.},

author = {Bombieri, Enrico, Masser, David, Zannier, Umberto},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

keywords = {Bounded Height Conjecture; Torsion Finiteness Conjecture},

language = {eng},

number = {1},

pages = {51-80},

publisher = {Scuola Normale Superiore, Pisa},

title = {Intersecting a plane with algebraic subgroups of multiplicative groups},

url = {http://eudml.org/doc/272261},

volume = {7},

year = {2008},

}

TY - JOUR

AU - Bombieri, Enrico

AU - Masser, David

AU - Zannier, Umberto

TI - Intersecting a plane with algebraic subgroups of multiplicative groups

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

PY - 2008

PB - Scuola Normale Superiore, Pisa

VL - 7

IS - 1

SP - 51

EP - 80

AB - Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed elsewhere.

LA - eng

KW - Bounded Height Conjecture; Torsion Finiteness Conjecture

UR - http://eudml.org/doc/272261

ER -

## References

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- [7] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - structure theorems and applications, Int. Math. Res. Not. IMRN 19 (2007), 33 pages. Zbl1145.11049MR2359537
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- [10] J. W. S. Cassels, “An Introduction to Diophantine Approximation”, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 45, Cambridge, 1965. Zbl0077.04801MR120219
- [11] T. Loher and D. Masser, Uniformly counting points of bounded height, Acta Arith.111 (2004), 277–297. Zbl1084.11034MR2039627
- [12] R. Pink, A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, manuscript dated 17th April 2005.
- [13] A. Schinzel, “Polynomials with Special Regard to Reducibility”, Encyclopaedia of Mathematics and its Applications, Vol. 77, Cambridge, 2000. Zbl0956.12001MR1770638
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- [15] B. Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc.65 (2002), 27–44. Zbl1030.11073MR1875133

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