  The values at positive integers of the polyzeta functions are solutions of
the polynomial equations arising from Drinfeld's associators, which have
numerous applications in quantum algebra. Considered as iterated integrals they
become periods of the motivic fundamental groupoid of
$\mathbb{P}^1\setminus\{0,1,\infty\}$. From there comes a fundamental, yet no
more explicit, system of algebraic relations; it implies the system of
associators.
  We focus here on the combinatorics properties of another system of relations,
the ``double shuffles'', which comes from elementary series and integrals
manipulations. We show that it shares a torsor property with associators and
``motivic'' relations, is implied by the latter and defines a polynomial
algebra over $\mathbb{Q}$ (a result first due to J. Ecalle). We obtain these
results for more general numbers: values of Goncharov's multiple polylogarithms
at roots of unity.
  These results were previously announced in math.QA:0012024. Here is the
detailed proof.
