Algorithms in Algebraic Geometry pp 133-152 | Cite as

# Solving Polynomial Systems Equation by Equation

## Abstract

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.

## Key words

Algebraic set component of solutions diagonal homotopy embedding equation-by-equation solver generic point homotopy continuation irreducible component numerical irreducible decomposition numerical algebraic geometry path following polynomial system witness point witness set## AMS(MOS) subject classifications

Primary 65H10 Secondary 13P05 14Q99 68W30## Preview

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