A. Esterov

The tools used to study polynomial equations with indeterminate coefficients are extended to some important cases with interrelated ones.

Methods for investigating polynomial equations with indeterminate coefficients cannot be applied to those with coefficients that are dependent on each other. This is the case if, for example, the equations are obtained by taking partial derivatives of another polynomial. Using tropical geometry, we were unexpectedly able to extend the classical methods to new classes of equations with interrelated coefficients.

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.

Engineered complete intersections: slightly degenerate Bernstein-Kouchnirenko-Khovanskii