Symbolic-numeric completion of differential systems by homotopy continuation
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
An intrinsic homotopy for intersecting algebraic varieties
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Application of numerical algebraic geometry and numerical linear algebra to PDE
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Numerical algebraic geometry and kinematics
Proceedings of the 2007 international workshop on Symbolic-numeric computation
An intrinsic homotopy for intersecting algebraic varieties
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Sampling algebraic sets in local intrinsic coordinates
Computers & Mathematics with Applications
Numerically Computing Real Points on Algebraic Sets
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Cell decomposition of almost smooth real algebraic surfaces
Numerical Algorithms
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We show how to use numerical continuation to compute the intersection $C=A\cap B$ of two algebraic sets A and B, where A, B, and C are numerically represented by witness sets. En route to this result, we first show how to find the irreducible decomposition of a system of polynomials restricted to an algebraic set. The intersection of components A and B then follows by considering the decomposition of the diagonal system of equations u - v = 0 restricted to {u,v}$\in$ A $\times$ B. An offshoot of this new approach is that one can solve a large system of equations by finding the solution components of its subsystems and then intersecting these. It also allows one to find the intersection of two components of the two polynomial systems, which is not possible with any previous numerical continuation approach.