Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations.
Homotopy continuation
The primary computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated solutions (points) of one are continued to the other. This is a specialization of the more general method of numerical continuation. Let z {\displaystyle z} represent the variables of the system. By abuse of notation, and to facilitate the spectrum of ambient spaces over which one can solve the system, we do not use vector notation for z {\displaystyle z} . Similarly for the polynomial systems f {\displaystyle f} and g {\displaystyle g} . Current canonical notation calls the start system g {\displaystyle g} , and the target system, i.e., the system to solve, f {\displaystyle f} . A very common homotopy, the straight-line homotopy, between f {\displaystyle f} and g {\displaystyle g} is
H ( z , t ) = ( 1 − t ) f ( z ) + t g ( z ) . {\displaystyle H(z,t)=(1-t)f(z)+tg(z).}
In the above homotopy, one starts the path variable at t start = 1 {\displaystyle t_{\text{start}}=1} and continues toward t end = 0 {\displaystyle t_{\text{end}}=0} . Another common choice is to run from 0 {\displaystyle 0} to 1 {\displaystyle 1} . In principle, the choice is completely arbitrary. In practice, regarding endgame methods for computing singular solutions using homotopy continuation, the target time being 0 {\displaystyle 0} can significantly ease analysis, so this perspective is here taken. Regardless of the choice of start and target times, the H {\displaystyle H} ought to be formulated such that H ( z , t start ) = g ( z ) {\displaystyle H(z,t_{\text{start}})=g(z)} , and H ( z , t end ) = f ( z ) {\displaystyle H(z,t_{\text{end}})=f(z)} . One has a choice in g ( z ) {\displaystyle g(z)} , including
Roots of unity Total degree Polyhedral Multi-homogeneous and beyond these, specific start systems that closely mirror the structure of f {\displaystyle f} may be formed for particular systems. The choice of start system impacts the computational time it takes to solve f {\displaystyle f} , in that those that are easy to formulate (such as total degree) tend to have higher numbers of paths to track, and those that take significant effort (such as the polyhedral method) are much sharper. There is currently no good way to predict which will lead to the quickest time to solve. Actual continuation is typically done using predictor–corrector methods, with additional features as implemented. Predicting is done using a standard ODE predictor method, such as Runge–Kutta, and correction often uses Newton–Raphson iteration. Because f {\displaystyle f} and g {\displaystyle g} are polynomial, homotopy continuation in this context is theoretically guaranteed to compute all solutions of f {\displaystyle f} , due to Bertini's theorem. However, this guarantee is not always achieved in practice, because of issues arising from limitations of the modern computer, most namely finite precision. That is, despite the strength of the probability-1 argument underlying this theory, without using a priori certified tracking methods, some paths may fail to track perfectly for various reasons.
Witness set A witness set
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