The preceding Etats de la recherche on this topic happened in Rennes, 2006, and the organizers of the present edition asked me to make the bridge between these two sessions. My 2006 talks were devoted to the theory of heights and equidistribution theorems for algebraic dynamical systems. I will start from there by presenting the framework allowed by Arakelov geometry, and explaining the recent manuscript of X. Yuan and S.-W. Zhang who provide a birational perspective to these concepts. The theory is a bit complex and technical but I will try to emphasize the parallel between those ideas and the ones that lie at the ground of pluripotential theory in complex analysis, or in the theory of b-divisors in algebraic geometry.[-]

Let $S$ be a multiplicatively defined set. Ostmann conjectured, that the set of primes cannot be (nontrivially) written as a sumset $P\sim A+B$ (even in an asymptotic sense, when finitely many deviations are allowed). The author had previously proved that there is no such ternary sumset $P\sim A+B+C$ (with $ \left |A \right |,\left |B \right |,\left |C \right |\geq 2$). More generally, in recent work we showed (with A. Harper) for certain multiplicatively defined sets $S$, namely those which can be treated by sieves, or those with some equidistribution condition of Bombieri-Vinogradov type, that again there is no (nontrivial) ternary decomposition $P\sim A+B+C$. As this covers the case of smooth numbers, this settles a conjecture of A.Sárközy.
Joint work with Adam J. Harper.[-]

Given a computer algebra problem described by polynomials with rational coefficients, I will present various tools that help measuring the cost of solving it, where cost means giving bounds for the degrees and heights (i.e. bit-sizes) of the output in terms of those of the input data. I will detail an arithmetic Bézout inequality and give some applications to zero-dimensional polynomial systems. I will also speak about the Nullstellensatz and Perron's theorem for implicitization, if time permits.[-]

Given a computer algebra problem described by polynomials with rational coefficients, I will present various tools that help measuring the cost of solving it, where cost means giving bounds for the degrees and heights (i.e. bit-sizes) of the output in terms of those of the input data. I will detail an arithmetic Bézout inequality and give some applications to zero-dimensional polynomial systems. I will also speak about the Nullstellensatz and Perron's theorem for implicitization, if time permits.[-]