Higher-Order Deflation for Polynomial Systems with Isolated Singular Solutions

Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton’s method. Using first-order derivatives of the polynomials in the system, our method creates an augmented system of equations which has the multiple isolated solution of the original system as a regular root. In this paper we consider two approaches to computing the “multiplicity structure” at a singular isolated solution. An idea coming from one of them gives rise to our new higher-order deflation method. Using higherorder partial derivatives of the original polynomials, the new algorithm reduces the multiplicity faster than our first method for systems which require several first-order deflation steps. 2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30.