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Polygonal lines are used for the paths of the gluon field phase factorsentering in the definition of gauge invariant quark Green's functions. Thisallows classification of the Green's functions according to the number ofsegments the polygonal lines contain. Functional relations are establishedbetween Green's functions with polygonal lines with different numbers ofsegments. An integrodifferential equation is obtained for the quark two-pointGreen's function with a path along a single straight line segment where thekernels are represented by a series of Wilson loop averages along polygonalcontours. The equation is exactly and analytically solved in the case oftwo-dimensional QCD in the large-$N_c$ limit. The solution displays generationof an infinite number of dynamical quark masses accompanied with branch pointsingularities that are stronger than simple poles. An approximation scheme,based on the counting of functional derivatives of Wilson loops, is proposedfor the resolution of the equation in four dimensions.

Two-point gauge invariant quark Green's functions with polygonal phase factor lines