Abstract

We show that the dimension of an algebraic (affine or projective) variety can be computed by a well parallelizable arithmetical network in non-uniform polynomial sequential time in the size of the input. This input is given by a system of polynomial equations written in dense representation. The coordinates of the ambient space can be put in Noether position with respect to the variety within the same time bounds. By the way, we consider as an intermediate problem the determination of the isolated points of the given variety, which is of obvious practical interest. We suppose that the base domain, from where the coefficients of the input polynomials are taken, is infinite and, in the case of an affine variety, that its field of fractions is perfect. If this domain consists of the integers, our algorithms can be realized by boolean networks of the same complexity type (however these networks are not uniform with respect to the number of variables occuring in the input polynomials). Ou...

Citations