Solving Polynomial Systems Equation by Equation

By a numerical continuation method called a diagonal homotopy, one can compute the intersection of two irreducible positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets that are not necessarily irreducible or even equidimensional. Of particular interest is the special case where one of the sets is defined by a single polynomial equation. This leads to an algorithm for finding a numerical representation of the solution set of a system of polynomial equations introducing the equations one by one. Preliminary computational experiments show this approach can exploit the special structure of a polynomial system, which improves the performance of the path following algorithms.

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