Solving nonlinear equations from higher order derivations in linear stages

Abstract

for all polynomials a ∈ R of degree less than or equal to d and all higher order derivations Eγ = Eγs · · ·Eγ1 and Fγ = Fγs · · ·Fγ1 , where γ = (γ1, . . . , γs), and 1 ≤ γi ≤M , of length s less or equal to r. The reason for wanting such derivations E1, . . . , EM is simple. Given several derivations F1, . . . , FM , it is useful to have derivations E1, . . . , EM which are good local approximations to the Fi and which are easy to compute with. Notice that since the right hand sides of Equations (∗) are known, while the left hand sides involve the unspecified coefficients of the polynomial functions, the equations are equivalent to a system of nonlinear algebraic equations involving the coefficients. It is the purpose of this paper to give an algorithm which solves such a system, when a solution exists. In fact, as the title suggests, this algorithm will work by solving a sequence of r linear systems.