Any working mathematician, denoted by F, is likely to come across systems of $n$ nonlinear analytic equations $f(x)=0$ in $n$ unknowns $x$.

The desires of F and the structure of $f$ determine the nature of possible output for a solution $x$.

In this talk we overview various scenarios where F considers $x$ that are regular or singular isolated solutions, and desires to

- approximate or rigorously isolate $x$

- find all solutions or just one solution

- do the above very fast or without much care for practical computational cost.

To illustrate, we use applications ranging from Schubert calculus, to celestial orbit estimation, to 3D reconstruction in computer vision.

(Based on joint work with F, where F ranges over a large set.)