Lower estimates of polynomials and linear forms in the values of $F$-series

The paper applies a modification of the generalized Sigel – Shidlovscii's method to values of $F$ – series at sufficiently small $p$ – adic points for a given $p$. The generalized Siegel – Shidlovskii's method is considerably developed in works by Chirskii V. G., Bertrand D., Yebbou Y, Matala–Aho T., Zudilin V. V., Matveev V. Yu., Andre Y. et al. But these papers dealt with the so called global relations and related notions such as infinite linear and algebraic independence. Here we consider values at points from a given field $\mathbb{Q}_p$. The notion of the infinite algebraic independence is related to a direct product of infinite set of fields $\mathbb{Q}_p$, it means that if $\alpha_1, \ldots, \alpha_n$ – are elements of this direct product with coordinates $\alpha_1^{(p)}, \ldots, \alpha_n^{(p)}$ in the field $\mathbb{Q}_p$, then for any non–zero polynomial with integer coefficients there exist infinitely many primes $p$ such that in $\mathbb{Q}_p$ one has $P(\alpha_1^{(p)}, \ldots, \alpha_n^{(p)})\neq 0$. But these results give no information for a specific $p$. Here we prove that a non–zero linear form and a non–zero polynomial do not vanish at values of the considered series at $p$ – adic points which are small enough, depending on the height of a linear form or a polynomial and depending on the degree of the polynomial. The results of these paper will be applied to the values of generalized hypergeometric $F$ – series.