Analytic complexity of functions of two variablesстатья
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Дата последнего поиска статьи во внешних источниках: 7 декабря 2013 г.
Аннотация:The definition of analytic complexity of an analytic function of two variables
is given. It is proved that the class of functions of a chosen complexity is a differentialalgebraic
set. A differential polynomial defining the functions of first class is constructed.
An algorithm for obtaining relations defining an arbitrary class is described. Examples of
functions are given whose order of complexity is equal to zero, one, two, and infinity. It is
shown that the formal order of complexity of the Cardano and Ferrari formulas is significantly
higher than their analytic complexity. The complexity classes turn out to be invariant with
respect to a certain infinite-dimensional transformation pseudogroup. In this connection, we
describe the orbits of the action of this pseudogroup in the jets of orders one, two, and three.
The notion of complexity order is extended to plane (or “planar”) 3-webs. It is discovered
that webs of complexity order one are the hexagonal webs. Some problems are posed.