An effective algorithm for deciding the solvability of a system of polynomial equations over 𝑝-adic integers

Consider a system of polynomial equations in n n variables of degrees at most d d with integer coefficients whose lengths are at most M M . By using a construction close to smooth stratification of algebraic varieties, it is shown that one can construct a positive integer Δ > 2 M ( n d ) c 2 n n 3 \begin{equation*} \Delta > 2^{M(nd)^{c\, 2^n n^3}} \end{equation*} (here c > 0 c>0 is a constant) depending on these polynomials and having the following property. For every prime p p the system under study has a solution in the ring of p p -adic numbers if and only if it has a solution modulo p N p^N for the least integer N N such that p N p^N does not divide Δ \Delta . This improves the previously known, at present classical result by B. J. Birch and K. McCann.