Systems of algebraic equations with bad reduction (Q1264468)

The author proposes a method for finding solutions of a system of algebraic equations with integer coefficients and finitely many solutions. The method is based on, given a prime number \(p\), and \(p\)-adic reduction of the system and the application of the Newton-Hensel method to lift a solution modulo \(p^k\) for greater values of \(k\) to find, afterwards, an algebraic number approximated by them. This method is well-known but has some problems: for example, not every solution of the reduced system in the ring of \(p\)-adic integers can be found or the finding of a solution of a system in a field with \(p\) elements can take a lot of time. The author proposes some remedies of these two difficulties in his approach, widening the scope of applicability of \(p\)-adic approximation. His algorithms solve some problems for `bad reduction' primes (i.e. primes for which there is no lift or there are more than one) by making a suitable change in the polynomials defining the original system. This allows him to attack some big systems successfully. He also gives details of the implementation of his procedures in MAPLE and a list of examples and timings.