Counting Affine Roots of Polynomial Systems via Pointed Newton Polytopes

Abstract

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 Bé zout 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.

The BKK root count in $\mathbf {C}^n$

“…Furthermore, our bound appears to be easier to compute. This bound also holds over arbitrary algebraically closed fields [10].…”

“…When 0 ∈ A i for some i, we will show in §2 that our bound is exact when the coefficients of the p i 's are chosen generically and the zeros of P (x) at infinity are nonsingular . An alternative combinatorial criterion to guarantee the exactness of this bound is given in [10]. In contrast to the complex analytic approach given in this paper, the method in [10] is more algebraic and the conditions are formulated in a more general setting.…”

“…An alternative combinatorial criterion to guarantee the exactness of this bound is given in [10]. In contrast to the complex analytic approach given in this paper, the method in [10] is more algebraic and the conditions are formulated in a more general setting.…”

Solving Degenerate Sparse Polynomial Systems Faster

“…Better still, we can read this off directly from our u-resultant by computing ( Our next main theorem tells us exactly how and when we can use a twisted Chow form to compute monomials in the roots of F . Recall that to any n-dimensional rational polytope Q ⊂ R n one can associate its corresponding toric variety (over K) T (Q) (Kempf et al, 1973;Danilov, 1978;Kapranov et al, 1992;Fulton, 1993;Gel'fand et al, 1994;Rojas, 1999a), and this T (Q) always has † a naturally embedded copy of (K * ) n . To state our results fully, we will require some toric variety terminology, but the underlying idea is simple: by working in compactifications more general than the projective spaces {P…”

Deformation Techniques for Sparse Systems

“…The complexity of such algorithms is usually determined by geometric invariants associated to the family of systems under consideration (see, e.g., [16], [25], [53], [44], [24], [27], [13], [50], [31], [39]), typically in the form of a suitable (arithmetic or geometric) Bézout number (see [36], [25], [33], [46], [26], [23], [43]). …”