We have considered paths obtained by the homotopy continuation and quadratic turning points. Now, we will suggest the way to approximate the quadratic turning point by using the method called the deflation method.
First, consider the following situation.
The homotopy path in is denoted by . is the quadratic turning point that we want to find and is the point that will be used to approximate by using Newton’s method. In order to use the Newton’s method, we need to consider the following polynomial system obtained by the property of .
The first condition is obtained since the quadratic turning point is the point on the path. The second condition is given by the property that the quadratic turning point is singular.
However, in here we encounter the problem about ‘inefficiency’. We know that is an matrix with polynomial entries. Therefore, computing the determinant of is a quite expensive task. We will consider the other way to use Newton’s method to to approximate . The point is making the system of polynomial which consists of much simpler polynomials. For simpler polynomials, we will burrow unknowns from the higher dimension space . Then, we can consider the following system with variables and $lates 2n+1$ equations.
where is the random vector in .
The first condition is trivial since the quadratic turning point is singular. The second condition is the variation of the condition . Since we have , we know that is a singular matrix, and hence, there is a nonzero element in the kernel of (actually, the dimension of the kernel of is i.e. the line in ). We consider as a coordinate of an arbitrary element of the kernel of . Then, we have . The last condition is the procedure to choose the only one vector from the kernel of . Since the kernel of is 1-dimensional, there are infinitely many elements in the kernel. Therefore, by choosing an arbitrary vector , we have the only one element which satisfies the condition of . It can be considered as the procedure to find the intersection between the kernel (the line) and (the plane).
Then, by using , we can approximate by Newton’s method. After we get , we get the point by just dropping coordinates.
Here is the additional question about this approach. How the ‘good’ value of can be chosen? It may be a problem to find the proper initial condition for the system because by the initial condition , the approximate value of can be changed substantially. Thus, it is needed to know how to decide .