Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1211–1244 | Cite as

An arithmetic Bernštein–Kušnirenko inequality

  • César Martínez
  • Martín SombraEmail author


We present an upper bound for the height of the isolated zeros in the torus of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bernštein–Kušnirenko theorem. Its proof is based on arithmetic intersection theory on toric varieties.


Height of points Laurent polynomials Mixed integrals Toric varieties \({\varvec{u}}\)-Resultants 

Mathematics Subject Classification

Primary 14G40 Secondary 14C17 14M25 52A41 



We thank José Ignacio Burgos, Roberto Gualdi and Patrice Philippon for useful discussions. We also thank Walter Gubler, Philipp Habegger and the referee for their helpful comments and suggestions for improvement on a previous version of this paper. Part of this work was done while the authors met at the Universitat de Barcelona and at the Université de Caen. We thank both institutions for their hospitality.


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Copyright information

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Authors and Affiliations

  1. 1.Laboratoire de mathématiques Nicolas Oresme, CNRS UMR 6139Université de CaenCaen CedexFrance
  2. 2.ICREABarcelonaSpain
  3. 3.Departament de Matemàtiques i InformàticaUniversitat de BarcelonaBarcelonaSpain
  4. 4.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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