Computer aided design and motion control lead to algebraic systems. Resultants of
multivariate polynomials are useful to solve such problems. Following Gelfand,
Kapranov and Zelevinsky, we calculate them via the Cayley Formula as determinant
of a complex formed by global sections of sheaves. These arise from the Koszul
complex generated by the polynomials, which we twist by a reflexive rank one bundle
corresponding to the shift of Newton polytopes by a rational vector, introduced by
Canny and Emiris. Again, inspired by these authors, we apply tight mixed
subdivisions of the polytopes to obtain regular minors of the differentials
required to evaluate the Cayley formula. Besides the assumption that the
Minkowski sum of all Newton polytopes in the system should be full dimensional,
there are no further constraints on the set of exponents defining the input
polynomials with indeterminate coefficients. Consequently, our resultant
coincide with the definition of D’Andrea and Sombra. This complements the
package
SparseResultant implemented by Staglianò (2021) which requires
stricter assumptions, including that each individual Newton polytope must be
full-dimensional.
