Solving deficient polynomial systems with homotopies which keep the subschemes at infinity invariant

By a deficient polynomial system of n polynomial equations in n unknowns we mean a system that has fewer solutions than that predicted by the total degree, or the Bézout number, of the system. If the system is mhomogeneous, the Bézout number can be considerably reduced. In this paper, we introduce a homotopy for numerically determining all isolated solutions of deficient m-homogeneous systems. The initial polynomial system Q is chosen which keeps the subschemes of H(x, t) = (1 t)aQ(x) + tP(x) at infinity invariant when / varies in [0, 1 ).