  The problem to be solved in this paper is to construct a general method of
proving whether a certain set is p-computable or not. The method is based on a
polynomial analogue of the classical Gandhi's fixed point theorem. The
classical Gandhi theorem uses the extension of the predicate with the help of
the special operator $\Gamma^{\Omega^*}_{\Phi(x)}$ whose smallest fixed point
is the $\Sigma$-set. The work uses a new type of operator -
$\Delta_0^p$-operator $\Gamma_{F_{P_1^{+}},...,F_{P_n^{+}}}^{\mathfrak{M}}$,
which extends predicates so that the smallest fixed point remains a
p-computable set. Moreover, if in the classical Gandhi's fixed point theorem
the special $\Sigma$-formula $\Phi(\overline {x})$ is used in the construction
of the operator, then in the new operator, instead of a single formula, special
generating families of formulas $F_ {P_1 ^ {+}},...,F_{P_n^{+}}$. This work
opens up broad prospects for the application of the polynomial analogue of the
Gandhi theorem in the construction of new types of terms and formulas, in the
construction of new data types and programs of polynomial computational
complexity in Turing complete languages.
