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

Abstract:

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