Symbolic homotopy construction

Abstract

The classical Theorem of Bézout yields an upper bound for the number of finite solutions to a given polynomial system, but is very often too large to be useful for the construction of a start system, for the solution of a polynomial system by means of homotopy continuation. The BKK bound gives a much lower upper bound for the number of solutions, but unfortunately, constructing a start system based on this bound seems as hard as solving the original given polynomial system. This paper presents a way for computing an upper bound together with the construction of a start system. The first computation is performed symbolically. Due to this symbolic computation, the constructed start system can be solved numerically more efficiently. The paper generalizes current approaches for homotopy construction towards the BKK bound.