On the exponent of geometric unipotent radicals of pseudo-reductive groups

Let k^′/ k be a finite purely inseparable field extension and let G^′ be a reductive k^′-group. We denote by G=Rk′/k(G′), the Weil restriction of G^′ across k^′/ k, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical R_u(G_k¯) in terms of invariants of the extension k^′/ k, starting with the case G′=GLn and applying these results to the case where G^′ is a simple group.