Geodesic
Systems with parameters, or efficiently solving systems of polynomial equations: 33~years later.~I
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVIII, Tome 462 (2017), pp. 122-166

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Consider a system of polynomial equations with parametric coefficients over an arbitrary ground field. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the solutions of the system are given by algebraic formulas depending only on this stratum. Each stratum is a quasiprojective algebraic variety with degree bounded from above by a subexponential function in the size of the input data. Also, the number of strata is subexponential in the size of the input data. Thus, here we avoid double exponential upper bounds on the degrees and solve a long-standing problem. 
@article{ZNSL_2017_462_a7,
     author = {A. L. Chistov},
     title = {Systems with parameters, or efficiently solving systems of polynomial equations: 33~years {later.~I}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {122--166},
     publisher = {mathdoc},
     volume = {462},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_462_a7/}
}
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A. L. Chistov. Systems with parameters, or efficiently solving systems of polynomial equations: 33~years later.~I. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVIII, Tome 462 (2017), pp. 122-166. http://geodesic.mathdoc.fr/item/ZNSL_2017_462_a7/