Functional variant of Polynomial Analogue of Gandy's Fixed Point Theorem

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.