  In this work, a functional variant of the polynomial analogue of the
classical Gandy's fixed point theorem is obtained. Sufficient conditions have
been found to ensure that the complexity of the recursive function does not go
beyond the polynomial. In 2021, we proved a polynomial analogue of the
classical Gandy's fixed point theorem. This became an important impetus for the
construction of p-complete programming languages. And such a language was first
built by us in 2022. The main result of that work was: a solution of the
problem P=L. Next are the followers of the works on building a new high-level
language and the idea of building a general programming methodology. But there
was one gap in our research: classes of recursive functions whose complexity
was polynomial were not described. In this work we found sufficient conditions
for such functions. In many ways, the main ideas of this work are similar to
the ideas that we used in the proof of the polynomial analogue of Gandy's fixed
point theorem. But there are also striking differences. Functions, as such,
differ quite strongly from predicates precisely in the multitude of their
values. If a predicate is either true or false, then a function can generally
take on a variety of values. Moreover, even if there are not many values, but
there is recursion and simple multiplication, then powers and factorials may
arise during the calculations, which, of course, can violate the polynomial
computational complexity of this function. Therefore, finding these
restrictions on recursive functions that would be soft enough for the class of
functions to be large, and at the same time tough enough not to go beyond
polynomiality, has been a problem for us for the last 3 years, after the proof
of the polynomial analogue Gandy's fixed point theorem in the case of predicate
extensions.
