We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the class of sequentially polynomial-time problems. Our algorithm can be extended to compute in parallel one irreducible complex factor of a multivariate integral polynomial. However, the coefficients of the computed factor are only represented modulo a not necessarily irreducible polynomial specifying a splitting field. A consequence of our algoithm is that multivariate polynomials over finite fields can be tested for absolute irreducibility in deterministic sequential polynomial time in the size of the input. We also obtain a sharp bound for the last prime p for which, when taking an absolutely irreducible integral polynomial modulo p, the polynomial's irreducibility in the algebraic closure of the finite field order p is not preserved.

. We give a new algorithm for resolving singularities of plane curves. The algorithm is polynomial time in the bit complexity model, does not require factorization, and works over Q or finite fields. 1 Introduction Resolving singularities is a central problem in computational algebraic geometry. In this paper we describe a new algorithm for resolving singularities of irreducible plane curves. The algorithm runs in polynomialtime in the bit complexity model, does not require polynomial factorization, and works over Q or any finite field. Classical algorithms for resolving singularities [2, 15, 7] use a combination of methods involving -- the Newton polygon, a polygon in Z 2 whose vertices are the exponents of terms in f ; -- Puiseux series, power series with fractional exponents. These algorithms take polynomial time if we assume efficient factorization over algebraic extensions of the base field and unit-time arithmetic these extensions. Teitelbaum [13] establishes bounds on the d...