$u$-generation: solving systems of polynomials equation-by-equation

We develop a new method that improves the efficiency of equation-by-equation algorithms for solving polynomial systems. Our method is based on a novel geometric construction, and reduces the total number of homotopy paths that must be numerically continued. These improvements may be applied to the basic algorithms of numerical algebraic geometry in the settings of both projective and multiprojective varieties. Our computational experiments demonstrate significant savings obtained on several benchmark systems. We also present an extended case study on maximum likelihood estimation for rank-constrained symmetric $n\times n$ matrices, in which multiprojective $u$-generation allows us to complete the list of ML degrees for $n\le 6.$