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  "body": "\n## Summary\n\nHomotopy continuation for polynomial systems computes all isolated solutions of a system of polynomial equations in several complex variables.\nIt deforms a start system, whose solutions are known, into the target system, and it follows each start solution along the deformation by [[numerical continuation|5V3GJ84CPQ]].\nIt is the basic algorithm of numerical algebraic geometry, and it is used for systems from kinematics, chemistry and other applications.\n\n## Explanations\n\n### L1 human\n\nSolving many polynomial equations at once is hard, but some systems are easy, such as $x^2 = 1$, $y^3 = 1$.\nHomotopy continuation slides the easy system into the hard one step by step.\nEach solution of the easy system moves along a path as the equations change, and where the paths end are the solutions of the hard system.\nFor a system of two equations of degrees $2$ and $3$, one follows $2 \\cdot 3 = 6$ paths, and these endpoints include every isolated solution.\n",
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