Symbolic Homotopy Construction
Jan Verschelde and Ronald Cools
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.
Key words. Bézout number, BKK bound, homotopy continuation
Applicable Algebra in Engineering, Communication and Computing
4(3):169-183, 1993.