A randomized Newton-Hensel lifting algorithm computes the unique representation f of a polynomial h in a subring generated by algebraically independent polynomials g_1,...,g_n in O~((nL_1+n^4+L_2)M(Delta,n)) field operations.
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Computing Polynomial Representation in Subrings of Multivariate Polynomial Rings
A randomized Newton-Hensel lifting algorithm computes the unique representation f of a polynomial h in a subring generated by algebraically independent polynomials g_1,...,g_n in O~((nL_1+n^4+L_2)M(Delta,n)) field operations.