Given a dominant rational self-map on projective variety over number field, we can define the arithmetic degree at point. It is known that any point less than or equal to first dynamical degree. In this paper, show there are densely many Q‾-rational points with maximal (i.e., whose degree) for self-morphisms varieties. For unirational varieties and Abelian varieties, sufficiently large field. W...

We generalize our methodology for computing with Zariski dense subgroups of SL(n,Z) and Sp(n,Z), to accommodate input H SL(n,Q) Sp(n,Q). A key task, backgrounded by the Strong Approximation theorem, is a minimal congruence overgroup H. Once we have this overgroup, may describe all quotients The case n=2 receives particular attention.

For a compact HyperKähler manifold X , we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a Lagrangian fibration.

We establish an extension of the Hopf–Tsuji–Sullivan dichotomy to any Zariski dense discrete subgroup a semisimple real algebraic group $ G $. then apply this Anosov subgroups $, which surprisingly presents different phenomenon depending on rank ambient

Zariski Problem (Cancellation of indeterminates) is settled affirmatively, that is, it is proved that : Let k be an algebraically closed field of characteristic zero and let n, m ∈ N. If R[Y1, . . . , Ym] ∼=k k[X1, . . . , Xn+m] as k-algebras, where Y1, . . . , Ym, X1, . . . , Xn+m are indetermoinates, then R ∼=k k[X1, . . . , Xn]. Zariski Problem is the following : Zariski Problem. Let k be an...

We give explicit versions of Helfgott’s Growth Theorem for SL2, as well as of the Bourgain-Gamburd argument for expansion of Cayley graphs modulo primes of subgroups of SL2(Z) which are Zariski-dense in SL2.