Abstract: In this paper, we study the properties of the zero set of a homotopy H:I^m×[0, 1]→ R^m and its piecewise linear approximation φ_δi:I^m×[0, 1]→R^m, These properties are very important for the homotopy simplex pivot algorithm. However, we prove that for almost every polynomial mapping the zero set of linear homotopy H(z, t)=tp(z)+(1-t)Q(z) consists of q=∏_j=1ⁿq_j disjoint differential curves, and the zero set of its piecewise linear approximation φ_δi, consists of some broken lines. Where δ_i→0, these broken lines tend to differential curves in the zero set of H.