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Given a separably closed field 
$K$ of characteristic 
$p>0$ and finite degree of imperfection, we study the 
$\sharp$ functor which takes a semiabelian variety 
$G$ over 
$K$ to the maximal divisible subgroup of 
$G(K)$. Our main result is an example where 
$G^{\sharp }$, as a ‘type-definable group’ in 
$K$, does not have ‘relative Morley rank’, yielding a counterexample to a claim in Hrushovski [J. Amer. Math. Soc. 9 (1996), 667–690]. Our methods involve studying the question of the preservation of exact sequences by the 
$\sharp$ functor, and relating this to issues of descent as well as model-theoretic properties of 
$G^{\sharp }$. We mention some characteristic 0 analogues of these ‘exactness-descent’ results, where differential algebraic methods are more prominent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, which is interesting in its own right, as well as providing a uniform treatment of the characteristic 0 and characteristic 
$p$ cases of ‘exactness descent’.
$\mathbb{C}$ AND ZILBER’S EXPONENTIAL FIELDS: ZERO SETS OF EXPONENTIAL POLYNOMIALS
We continue the research programme of comparing the complex exponential with Zilberś exponential. For the latter, we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for 
$\mathbb{C}$ using analytic function theory, for example, the Identity Theorem.
It is well known that a finitely generated group 
${\rm\Gamma}$ has Kazhdan’s property (T) if and only if the Laplacian element 
${\rm\Delta}$ in 
$\mathbb{R}[{\rm\Gamma}]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in 
$\mathbb{R}[{\rm\Gamma}]$. Namely, 
${\rm\Gamma}$ has property (T) if and only if there exist a constant 
${\it\kappa}>0$ and a finite sequence 
${\it\xi}_{1},\ldots ,{\it\xi}_{n}$ in 
$\mathbb{R}[{\rm\Gamma}]$ such that 
${\rm\Delta}^{2}-{\it\kappa}{\rm\Delta}=\sum _{i}{\it\xi}_{i}^{\ast }{\it\xi}_{i}$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in 
$\mathbb{R}[{\rm\Gamma}]$, possibly with the assistance of computers.
We consider systems of quasi-periodic linear differential equations associated to a ‘resonant’ frequency vector 
${\it\omega}$, namely, a vector whose coordinates are not linearly independent over 
$\mathbb{Z}$. We give sufficient conditions that ensure that a small analytic perturbation of a constant system is analytically conjugate to a ‘resonant cocycle’. We also apply our results to the non-resonant case: we obtain sufficient conditions for reducibility.