Effective Noether irreducibility forms and applications

An absolutely irreducible multivariate polynomial is a polynomial that cannot be factored even if the coefficients of the factors can Yle in any field above the original coefficient field. For example, an absolutely irreducible polyn~ mial with rational number coefficients does not factor over the field of complex numbers. We consider algorithms that factor a multivariate polynomial into absolutely irreducible polynomials. Our contributions are essentially threefold. First, we estimate the blt complexity of our “generic” algorithms, which are defined for algebraic extensions of an arbitrary coefficient field, so that the polynomial size growth estimates apply to any input field in which elements are canonical y represented. The generic analysis also yields several effective irreducibility y theorems, among them a new effective Hilbert irreducibility theorem, which reduces the number of random bits previously needed when factoring polynomials in many variables. Finally, we introduce the “single path lazy factorization” representation model for elements in algebraic extension fields, with which the problem of factoring polynomials with rational number or rational function coefficients can be solved within the parallel computational complexity class MC. As a consequence, arbitrary high precision complex rational approximations or, in the function field case, truncated power series approximations, of the factor coefficients can be computed from our representation, again within NC complexity. One of our irreducibility y theorems also gives an upper bound on the accuracy needed to keep the approximations of the absolutely irreducible factors themselves absolutely irreducible.