General systems of linear forms: Equidistribution and true complexity

In this work, we give a complete characterization for these distributions for arbitrary systems of linear forms. This extends previous works which accomplished this in some special cases. As an application, we resolve a conjecture of Gowers and Wolf on the true complexity of linear systems. Our proof deviates from that of the previously known special cases and requires several new ingredients. One of which, which may be of independent interest, is a new theory of homogeneous nonclassical polynomials.