Solving Nonlinear Polynomial Systems via Symbolic-Numeric Elimination Method

Consider a general polynomial system S in x1, . . . , xn of degree q and its corresponding vector of monomials of degree less than or equal to q. The system can be written as M0 · [xq1, xq−1 1 x2, . . . , xn, x1, . . . , xn, 1] = [0, 0, . . . , 0, 0, . . . , 0, 0] (1) in terms of its coefficient matrix M0. Here and hereafter, [...] T means the transposition. Further, [ξ1, ξ2, . . . , ξn] is one of the solutions of the polynomial system, if and only if [ξ 1 , ξ q−1 1 ξ2, . . . , ξ 2 n, ξ1, . . . , ξn, 1] T (2) is a null vector of the coefficient matrix M0. Since the number of monomials is usually bigger than the number of polynomials, the dimension of the null space can be big. The aim of completion methods, such as ours and those based on Gröbner bases and others [4, 5, 6, 7, 8, 10, 16, 18, 17, 12, 9, 20], is to include additional polynomials belonging to the ideal generated by S, to reduce the dimension to its minima. The bijection φ : xi ↔ ∂ ∂xi , 1 ≤ i ≤ n, (3) maps the system S to an equivalent system of linear homogeneous PDEs denoted by R. Jet space approaches are concerned with the study of the jet variety V (R) = {( u q , u q−1 , . . . , u 1 , u ) ∈ J : R ( u q , u q−1 , . . . , u 1 , u ) = 0 } , (4) where u j denotes the formal jet coordinates corresponding to derivatives of order exactly j. A single prolongation of a system R of order q consists of augmenting the system with all possible derivatives of its equations, so that the resulting augmented systems, denoted by DR, has order q + 1. Under the bijection φ, the equivalent operation for polynomial systems is to multiply by monomials, so that the resulting augmented system has degree q + 1. A single geometric projection is defined as E(R) := {( u q−1 , . . . , u 1 , u ) ∈ Jq−1 : ∃ u q , R ( u q , u q−1 , . . . , u 1 , u ) = 0 } . (5)