# EliminationMatrices -- resultants

## Description

EliminationMatrices is a package for elimination theory, emphasizing universal formulas, in particular, resultant computations.

The package contains an implementation for computing determinant of free graded complexes, called detComplex, with several derived methods: listDetComplex, mapsComplex, and minorsComplex. This provides a method for producing universal formulas for any family of schemes, just by combining the resolution(Ideal) method with detComplex. In Section 2 determinants of free resolutions are treated, as well as a few examples. We recommend to see [Dem84, Jou95, GKZ94, Bus06] for more details on determinants of complexes in elimination theory.

The package also provides a method eliminationMatrix for computing matrices and formulas for different resultants applicable on different families of polynomials, such as the Macaulay resultant (Macaulay) for generic homogeneous polynomials; the residual resultant ('ciResidual' and 'CM2Residual') for generic polynomials having a non empty base locus scheme; the determinantal resultant ('determinantal') for generic polynomial matrices of a given generic rank. For the theory behind those resultants, the reader can refer to [Mac02, Jou91, Cha93, GKZ94, Jou97, CLO98, BEM00, BEM01, Bus01b, Bus06, Bus04].

The goal of this package is to provide universal formulas for elimination. The main advantage of this approach consists in the fact that one can provide formulas for some families of polynomials just by taking determinant to a free resolution. A direct consequence of a universal formula is that it is preserved by base change, in particular it commutes with specialization. A deep study of universal formulas for the image of a map of schemes can be seen in [EH00].

Bibliography:

[BBD12] Nicolás Botbol, Laurent Busé and Manuel Dubinsky. PDF. Package for elimination theory in Macaulay2 (2012).

[BEM00] Laurent Busé, Mohamed Elkadi and Bernard Mourrain, Generalized resultants over unirational algebraic varieties, J. Symbolic Comput. 29 (2000), no. 4-5, 515–526.

[BEM01] Laurent Busé, Mohamed Elkadi and Bernard Mourrain, Resultant over the residual of a complete intersection, Journal of Pure and Applied Algebra 164 (2001), no. 1-2, 35–57.

[Bus01a] Laurent Busé, Residual resultant over the projective plane and the implicitization problem, International Symposium on Symbolic and Algebraic Computing (ISSAC), ACM, (2001). Please, see the errata.pdf attached file., pp. 48–55.

[Bus04] Laurent Busé, Resultants of determinantal varieties, J.Pure Appl. Algebra193 (2004), no.1-3, 71–97.

[Bus06] Laurent Busé, Elimination theory in codimension one and applications, (2006).

[Cha93] Marc Chardin, The resultant via a Koszul complex, Computational algebraic geometry (1992), Progr. Math, vol. 109, Birkhäuser Boston, Boston, MA, pp. 29–39.

[CLO98] David Cox, John Little and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, (1998).

[Dem94] Michel Demazure, Une définition constructive du resultant, Centre de Mathématiques de l’Ecole Polytechnique 2 (1984), no. Notes informelles du calcul formel 1984-1994, 0–23.

[EH00] David Eisenbud and Joe Harris, The geometry of schemes., Graduate Texts in Mathematics. 197. New York, NY: Springer. x, 294 p., (2000).

[GKZ94] Israel M. Gel′fand, Mikhail M. Kapranov and Andrei V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, (1994). Mathematics: Theory & Applications, Birkh ̈auser Boston Inc, Boston, MA.

[Jou91] Jean-Pierre Jouanolou, Le formalisme du résultant, Adv. Math 90 (1991), no. 2, 117–263.

[Jou97] Jean-Pierre Jouanolou, Formes d’inertie et résultant: un formulaire, Adv. Math. 126 (1997), no. 2, 119–250.

[Mac02] Francis S. Macaulay, Some formulae in elimination, Proc. London Math. Soc. 33 (1902), no. 1, 3–27.

## Version

This documentation describes version 1.4 of EliminationMatrices.

## Source code

The source code from which this documentation is derived is in the file EliminationMatrices.m2.

## Exports

• Functions and commands
• bezoutianMatrix -- returns a matrix associated to generalized resultants
• ciResDeg -- compute a regularity index and partial degrees of the residual resultant over a complete intersection
• ciResDegGH -- compute a regularity index used for the residual resultant over a complete intersection
• degHomPolMap -- return the base of monomials in a subset of variables, and the matrix of coefficients of a morphism of free modules f:R(d1)+...+R(dn)->R_d with respect to these variables
• detComplex -- This function calculates the determinant of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.
• detResDeg -- compute a regularity index and partial degrees of the determinantal resultant
• eliminationMatrix -- returns a matrix that represents the image of the map
• listDetComplex -- This function calculates the list with the determinants of some minors of the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.
• macaulayFormula -- returns two matrices such that the ratio of their determinants is the Macaulay resultant
• mapsComplex -- This function calculates the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.
• maxCol -- Returns a submatrix form by a maximal set of linear independent columns.
• maxMinor -- Returns a maximal minor of the matrix of full rank.
• minorsComplex -- calculate some minors of the maps of a graded ChainComplex in a subset of variables and fixed degree
• regularityVar -- computes the Castelnuovo-Mumford regularity of homogeneous ideals in terms of Betti numbers, with respect to some of the variables of the ring
• Methods
• "bezoutianMatrix(List,Matrix)" -- see bezoutianMatrix -- returns a matrix associated to generalized resultants
• "degHomPolMap(Matrix,List,List,ZZ)" -- see degHomPolMap -- return the base of monomials in a subset of variables, and the matrix of coefficients of a morphism of free modules f:R(d1)+...+R(dn)->R_d with respect to these variables
• "degHomPolMap(Matrix,List,ZZ)" -- see degHomPolMap -- return the base of monomials in a subset of variables, and the matrix of coefficients of a morphism of free modules f:R(d1)+...+R(dn)->R_d with respect to these variables
• "detComplex(ZZ,List,ChainComplex)" -- see detComplex -- This function calculates the determinant of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.
• eliminationMatrix(List,Matrix) -- returns a matrix associated to the Macaulay resultant
• eliminationMatrix(List,Matrix,Matrix) -- returns a matrix corresponding to a residual resultant
• eliminationMatrix(ZZ,List,Matrix) -- returns a matrix corresponding to the determinantal resultant, in particular the Macaulay resultant
• "listDetComplex(ZZ,List,ChainComplex)" -- see listDetComplex -- This function calculates the list with the determinants of some minors of the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.
• "macaulayFormula(List,Matrix)" -- see macaulayFormula -- returns two matrices such that the ratio of their determinants is the Macaulay resultant
• "mapsComplex(ZZ,List,ChainComplex)" -- see mapsComplex -- This function calculates the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.
• "maxCol(Matrix)" -- see maxCol -- Returns a submatrix form by a maximal set of linear independent columns.
• "maxMinor(Matrix)" -- see maxMinor -- Returns a maximal minor of the matrix of full rank.
• "minorsComplex(ZZ,List,ChainComplex)" -- see minorsComplex -- calculate some minors of the maps of a graded ChainComplex in a subset of variables and fixed degree
• "regularityVar(List,Ideal)" -- see regularityVar -- computes the Castelnuovo-Mumford regularity of homogeneous ideals in terms of Betti numbers, with respect to some of the variables of the ring
• Symbols
• byResolution -- Strategy for eliminationMatrix.
• ciResidual -- Strategy for eliminationMatrix.
• CM2Residual -- Strategy for eliminationMatrix.
• determinantal -- Strategy for eliminationMatrix.
• Exact -- Strategy for functions that uses rank computation.
• Macaulay -- Strategy for eliminationMatrix.
• Numeric -- Strategy for functions that uses rank computation.
• Sylvester -- Strategy for eliminationMatrix.

## For the programmer

The object EliminationMatrices is .