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 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 Delta < 2(M(nd)c2nn3) (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.