Determining singular solutions of polynomial systems via symbolic–numeric reduction to geometric involutive forms

We present a method based on symbolic–numeric reduction to geometric involutive form to compute the primary component of and a basis of Max Noether space for a polynomial system at an isolated singular solution. The singular solution can be known exactly or approximately. For the case where the singular solution is known with limited accuracy, we then propose a generalized quadratic Newton iteration for refining it to high accuracy.