Systems with parameters, or efficiently solving systems of polynomial equations 33 years later

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 union of strata. For values of 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 the 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. This solves a long standing problem to avoid double exponential growth of coefficients for this problem.