Quantifier elimination for approximate BK-factorization

Factorization of linear partial differential operators (LPDOs) is a very wellstudied problem and a lot of pure existence theorems are known. The only known constructive factorization algorithm Beals-Kartashova (BK) factorization is presented in [1]). Its comparison with Hensel descent which is sometimes regarded as constructive, is given in [2], where the idea to use BKfactorization for approximate factorization is also discussed. It originates in one of the most interesting features of BK-factorization: at the beginning all the first-order factors are constructed and afterwards the factorization condition(s) should be checked. This leads to the important application area namely, numerical simulations which could be simplified substantially if instead of computation with one LPDE of order n we will be able to proceed computations with n LPDEs all of order 1. In numerical simulations it is not necessary to fulfill factorization conditions exactly but with some given accuracy, which we call approximate factorization. The idea of the present paper is to look into the feasibility of solving problems of this kind using quantifier elinination by cylindrical algebraic decomposition [3]. In this paper we are going to apply this approach to a hyperbolic LPDO of order 2 with polynomial coefficients.