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\begin{document}

\centerline{\bf On free algebras}

\centerline{\it (joint work with I. Kashuba at USP and J. Germoni at UdL)}

\centerline{\it by Olivier Mathieu at UdL}

In this work, we define some abstract algebraic object. Conjecturally, we expect these object to decribe
free alternative and free Jordan algebras.

The conjecture has been checked in some cases.
For one generator free Jordan and alternative algebras,
the conjecture holds: this approach provides a complex proof of the quite trivial fact that the free Jordan/alternative algebra in one
variable is the one variable polynomial algebra.
Also for free algebras with $\geq 2$ generators,
some numerical consequences has been checked by computer.
For example, the conjecture is compatible with the existence Glennie identities (some degree 8 Jordan polynomial in three variables) and with the fact that tetrads are not Jordan polynomials.

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