  We consider the fragmentation equation $\dfrac{\partial}{\partial t}f (t, x)
= --B(x)f (t, x) + \int\_{ y=x}^{ y=\infty} k(y, x)B(y)f (t, y)dy,$ and address
the question of estimating the fragmentation parameters-i.e. the division rate
$B(x)$ and the fragmentation kernel $k(y, x)$-from measurements of the size
distribution $f (t, $\times$)$ at various times. This is a natural question for
any application where the sizes of the particles are measured experimentally
whereas the fragmentation rates are unknown, see for instance (Xue, Radford,
Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a
polynomial division rate $B(x) = \alpha x^{\gamma}$ and a self-similar
fragmentation kernel $k(y, x) = \frac{1}{y} k\_0 (x/ y)$, we use the asymptotic
behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to
obtain uniqueness of the triplet $(\alpha, \gamma, k \_0)$ and a representation
formula for $k\_0$. To invert this formula, one of the delicate points is to
prove that the Mellin transform of the asymptotic profile never vanishes, what
we do through the use of the Cauchy integral.
