  We consider a dynamics of a generic birational plane map \Phi_n: CP^2 \to
CP^2, CP^2 -image of the birational mapping (inverse map is also rational)F_n :
C^2 \to C^2 and its such important characteristic as the Arnold complexity
C_A(k), which is proportional d(k)=deg(\Phi_n^k)- a degree of k-iteration of
the map \Phi_n, on the basis on algebraic-geometrical properties of such maps.
Additional importance of this characteristic follows from the Veselov
conjecture about the polynomial boundedness of the growth of d(k) for
integrable dynamical systems with a discrete time defined by birational plane
maps. The autonomous linear difference equation with integer coefficients for
d(k) is obtained. This equation is fully defined by \sigma_1 nonnegative
integers m_1,..., m_\sigma_1 that are determined by relations:
\Phi_n^{-m_i}(O_\alpha_i)=O^(-1)_\beta_i, i\in(1,2,...,\sigma_1), where
\Phi_n^{-m_i} is m_i-iteration of inverse map, O_\alpha_i, O^(-1)_\beta_i,
\alpha_i, \beta_i \in (1,2,...,\sigma), are indeterminacy points of the direct
and inverse maps, \sigma_1\leq\sigma and \sigma is a number of indeterminacy
points of \Phi_n,\Phi_n^{-1}. If \sigma_1 is equal to zero that d(k)=n^k,
otherwise the growth of d(k) is fully defined by a root spectrum of the secular
equation associated with the difference equation for d(k). The Veselov
conjecture corresponds to the root spectrum consisting of values being equal to
modulo one. The author doesn't suppose that the reader has acquaintance with
the algebraic geometry (AG) in CP^2 and the dynamical systems theory (DST) or
the functional equations since in the paper there are given all needed
definitions of used concepts of AG or DST and theorems.
