ResultantComplexes

Description

Systems of sparse polynomials with indeterminate coefficients are defined by the family of their supports which contain the exponents of terms. The Minkowski sum of their Newton polytopes specifies a toric variety. To calculate resultants Canny and Emiris shifted this sum by a rational vector. This displacement can also be viewed as a twist of line bundles associated with lattice polytopes by a suitable rank one-bundle. This way, a complex of finitely generated modules over the ring of coefficients is obtained as proposed by Gelfand, Kapranov and Zelevinsky via global sections of twisted sheafs in the Koszul complex generated by the given system. Following Canny and Emiris again, tight mixed subdivisions are utilized to obtain regular minors of the corresponding differentials, so that the determinant can be calculated by means of the Cayley formula. Besides the assumption that the Minkowski sum of all Newton polytopes in the system must be full dimensional, there are no further conditions for the family of supports. Consequently, the determinant of the complex agrees with the resultant, redefined by D'Andrea and Sombra. The example illustrates a system that contains the first three supports as unique essential sub-family. It results from the system considered in resultantComplex by multiplying the exponents with a unimodular integer matrix. Furthermore, the length of the fourth support in the factor lattice over the sub-lattice generated by the first three supports doubles the multiplicity of the resultant, as described by D'Andrea, C. and Sombra.

i1 : A = QQ[c_1..c_10];

i2 : supp = {{{0,0,0},{0,2,4},{-2,5,8}}, {{-2,4,6},{1,0,1},{4,-4,-4}}, {{3,-3,-3},{0,1,2}}, {{0,0,0},{2,-4,-4}}};

i3 : Lift = {{0,-12,-4},{-2,-1},{-3,0},{-5,-2}};

i4 : (complexRes,mon,choice) = resultantComplex( A, supp, Lift, {-1/8,1/3,4/7} );

i5 : complexRes

      44      46      2
o5 = A   <-- A   <-- A  <-- 0
                             
     0       1       2      3

o5 : Complex

i6 : Res = factor CayleyFormula(complexRes,mon,choice)

         2               2 14
o6 = (c c  - c c c  + c c )
       4 7    5 7 8    6 8

o6 : Expression of class Product

References

Canny, John F. and Emiris, Ioannis Z., "A Subdivision-based Algorithm for the Sparse Resultant", Journal of ACM, May 2000, volume 47 number 3, 2000
D'Andrea, C. and Sombra, M., A Poisson formula for the sparse resultant., Proc. Lond. Math. Soc. (3), ISSN 0024-6115; 1460-244X/e, Volume 110, Number 4, Pages 932--964, 2015
Sturmfels, Bernd, "On the Newton Polytope of the Resultant" , Journal of Algebraic Combinatorics, 1994, Apr, volume 3, number 2, pages 207-236 ISSN 1572-9192
Gelfand, I.M. and Kapranov, M. and Zelevinsky, A., "Discriminants, Resultants, and Multidimensional Determinants", ISBN 9780817647711, Modern Birkhäuser Classics, 1994

resultantComplex -- Resultant complex of sparse system.
calcResultant -- Calculate resultant of sparse system.
CayleyFormula -- Calculate determinant of resultant complex.
randomLift -- Get random lift-vectors to produce coherent subdivisions.
polynomials -- convert support set of exponents to polynomial
mixedSubdivision -- computes a coherent mixed sub-division
CannyEmirisCoef -- determine rank one bundle to twist Koszul complex

Author

Friedemann Groh

Certification

Version 1.0 of this package was accepted for publication in volume 16 of Journal of Software for Algebra and Geometry on 2025-08-16, in the article Resultant complexes of toric systems (DOI: 10.2140/jsag.2026.16.1). That version can be obtained from the journal.

Version

This documentation describes version 1.0 of ResultantComplexes, released August 16, 2025.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ResultantComplexesSource,
  title = {{ResultantComplexes: A \emph{Macaulay2} package. Version~1.0}},
  author = {Friedemann Groh},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

@article{ResultantComplexesArticle,
  title = {{Resultant complexes of toric systems}},
  author = {Friedemann Groh},
  journal = {Journal of Software for Algebra and Geometry},
  volume = {16},
  year = {2025},
}

Exports

Functions and commands
- calcResultant -- Calculate resultant of sparse system.
- CannyEmirisCoef -- determine rank one bundle to twist Koszul complex
- CayleyFormula -- Calculate determinant of resultant complex.
- mixedSubdivision -- computes a coherent mixed sub-division
- polynomials -- convert support set of exponents to polynomial
- randomLift -- Get random lift-vectors to produce coherent subdivisions.
- resultantComplex -- Resultant complex of sparse system.
Methods
- calcResultant(List,List,List) -- see calcResultant -- Calculate resultant of sparse system.
- CannyEmirisCoef(Sequence,Matrix) (missing documentation)
- mixedSubdivision(List,Matrix) (missing documentation)
- polynomials(List,List,List) (missing documentation)
- randomLift(List,ZZ) (missing documentation)
- resultantComplex(List,List,List) -- see resultantComplex -- Resultant complex of sparse system.
- resultantComplex(PolynomialRing,List,List,List) -- see resultantComplex -- Resultant complex of sparse system.

For the programmer

The object ResultantComplexes is a package, defined in ResultantComplexes.m2.

The source of this document is in ResultantComplexes.m2:378:0.