A. Esterov

Investigating the geometry of a system of sparse polynomial equations, beyond the classical genericity assumption on their coefficients.

The results of Bernstein-Kouchnirenko-Khovanskii are used to study the geometry of polynomial equations. But these results are not applicable if, for example, the equations are obtained by taking partial derivatives or permuting the variables of another. We extend the classical results to a natural class of  ‘systems of equations that are nondegenerate upon cancellations’, which includes many such important cases. 

Geometry of sparse systems of polynomial equations (i.e. the ones with prescribed monomials and generic coefficients) is well studied in terms of their Newton polytopes. The results of this study are colloquially known as the Bernstein--Kouchnirenko--Khovanskii toolkit, and unfortunately are not applicable to many important systems, whose coefficients slightly fail to be generic. This for instance happens if some of the equations are obtained from another one by taking partial derivatives or permuting the variables, or the equations are linear, realizing a non-trivial matroid, or in more advanced settings such as generalized Calabi--Yau complete intersections. Such interesting examples (as well as many others) turn out to belong to a natural class of ``systems of equations that are nondegenerate upon cancellations''. We extend to this class several classical and folklore results of the Bernstein--Kouchnirenko--Khovanskii toolkit, such as the ones regarding the number and regularity of solutions, their irreducibility, tropicalization and Calabi--Yau-ness.

Slight degenerations

Engineered complete intersections: slightly degenerate Bernstein-Kouchnirenko-Khovanskii