• multiprojective geometry
• height
• transcendental number theory
• convex body

In the first part we recall the theory of multiprojective elimination initiated by P.Philippon and developed by G.Rémond. In particular, we define the eliminant ideal, the resultant forms and the Hilbert-Samuel polynomial for multigraded modules. We then look at subvarieties and cycles of a product of projective spaces, over a number field, and we define their mixed degrees and mixed heights, which measure respectively their geometric and arithmetic complexity. Finally, we define the heights of multiprojective cycles relative to some sets of polynomials, generalizing a previous notion of height due to M.Laurent and D.Roy, and we give detailed proofs for their properties.
In the second part we prove that if we have a sequence of polynomials with bounded degrees and bounded integer coefficients taking small values at a pair (a,b) together with their first derivatives, then both a and b need to be algebraic. The main ingredients of the proof include a translation of the problem in multihomogeneous setting, an interpolation result, the construction of a 0-dimensional variety with small height, a result for the multiplicity of resultant forms, and a final descent.
This work is motivated by an arithmetic statement equivalent to Schanuel's conjecture, due to D.Roy.