Efficient absolute factorization of polynomials with parametric coefficients

Consider a polynomial with parametric coefficients. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the decomposition of this polynomial into absolutely irreducible factors is given by algebraic formulas depending only on the stratum. Each stratum is a quasiprojective algebraic variety. This variety and the corresponding output are given by polynomials of degrees at most $D$ with $D=d'd^{O(1)}$ where $d',d$ are bounds on the degrees of the input polynomials. The number of strata is polynomial in the size of the input data. This solves a long-standing problem of avoiding a double exponential growth of the degrees of coefficients for this problem.