Solving Degenerate Sparse Polynomial Systems Faster

Details Solving Degenerate Sparse Polynomial Systems Faster Solving Degenerate Sparse Polynomial Systems Faster

1998-09-14

arxiv.org/abs/math/9809071v2

Mathematics

Algebraic Geometry

This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is

Scientific paper

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.

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