[Author]
 Chistov, Alexander L.

[Title]
 An Effective Algorithm for Deciding Solvability of a System of Polynomial
 Equations over $p$-adic Integers

[AMS Subj-class]
 11D88 $p$-adic and power series fields
 14Q15 Computational aspects of higher-dimensional varieties

[Keywords]
 $p$-adic integers, polynomial systems, decidability algorithm

[Abstract]
 Consider a system of polynomial equations in $n$ variables of degrees at most
 $d$ with integer coefficients with the  lengths at most $M$.  We show using
 the construction close to smooth stratification of algebraic varieties that
 one can construct a positive integer \[\Delta < 2^{M(nd)^{c\, 2^n n^3}}\]
 (here $c>0$ is a constant) depending on these polynomials and 
 satisfying the following property. For every prime $p$ the considered 
 system 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.

[Comments]
 LaTeX, English, 26 pp.
 This is a completely revised version of my old unpublished preprint on the same subject 
 previously written in the coauthorship with M. Karpinski. 
 I have strengthened the results and corrected many inacuracies. 
 Now I am going to publish it in the Journal "Algebra i Analiz", at first in Russian.

[Contact e-mail]
 alch@pdmi.ras.ru