Abstract : 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.

Keywords : arithmetic; Bézout inequality; nullstellensatz; shape lemma; Perron theorem