Counting Affine Roots of Polynomial Systems via Pointed Newton Polytopes

We give a new upper bound on the number of isolated roots of a polynomial system. Unlike many previous bounds, our bound can also be restricted to different open subsets of affine space. Our methods give significantly sharper bounds than the classical Bezout theorems and further generalize the mixed volume root counts discovered in the late 1970s. We also give a complete combinatorial classification of the subsets of coefficients whose genericity guarantees that our bound is actually an exact root count in affine space. Our results hold over any algebraically closed field.

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