This paper deals with the estimation of multiantenna channels in the line-of-sight conditions that are prevalent in the near field. By expressing the curved wavefront as a polynomial via a power series expansion of a sphere, the estimation of the channel over the array can be formulated as a multidimensional polynomial phase estimation problem. The application of a newly developed polynomial phase estimator, able of handling arbitrary dimensions and polynomial degrees, yields a superior tradeoff between channel estimation accuracy and complexity.