Solving Univariate P-adic Constraints

We describe an algorithm for solving systems of univariate p-adic constraints. In analogy with univariate real constraints, we formalize univariate p-adic constraints as univariate polynomial equations and order comparisons between p-adic values of univariate polynomials. Systems of constraints are arbitrary boolean combinations of such constraints. Our method combines techniques of Presburger arithmetic for integer p-adic values with a detailed analysis of the combined range of p-adic values of a list of univariate polynomials with rational coefficients and p-adic arguments. We describe an algorithmic test for the solvability of such constraints in the field Qp of p-adic numbers. If this test has a positive outcome, then we provide in addition to this information a sample solution of the constraint system, either as a rational number or as a rational approximation of a uniquely determined algebraic p-adic number. An implementation of our algorithm is under way based on the REDLOG package of REDUCE. First results obtained with this implementation are displayed. [email protected], http://www.fmi.uni-passau.de/ ̃sturm/ [email protected]