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# factorModuleBasis -- enumerate standard monomials

## Synopsis

• Usage:
F = factorModuleBasis(J)
• Inputs:
• Outputs:
• F, an instance of the type FactorModuleBasis, a partition of the set of monomials that are not leading monomial of any element of the module spanned by J, into monomial cones

## Description

The result represents a collection of finitely many cones of monomials, each cone being the set of multiples of a certain monomial by all monomials in certain variables; the generating monomials are accessed by basisElements; the sets of variables for each cone are obtained from multVar.

 i1 : R = QQ[x,y,z]; i2 : M = matrix {{x*y,x^3*z}}; 1 2 o2 : Matrix R <-- R i3 : J = janetBasis M; i4 : F = factorModuleBasis J +--+------+ o4 = |1 |{z, y}| +--+------+ |x |{z} | +--+------+ | 2| | |x |{z} | +--+------+ | 3| | |x |{x} | +--+------+ o4 : FactorModuleBasis i5 : basisElements F o5 = | 1 x x2 x3 | 1 4 o5 : Matrix R <-- R i6 : multVar F o6 = {set {z, y}, set {z}, set {z}, set {x}} o6 : List
 i7 : R = QQ[x,y]; i8 : M = matrix {{x*y-y^3, x*y^2, x*y-x}, {x, y^2, x}}; 2 3 o8 : Matrix R <-- R i9 : J = janetBasis M +--------------+------+ o9 = || y3-x | |{y} | || 0 | | | +--------------+------+ || xy-x | |{y} | || x | | | +--------------+------+ || x2y-x2 | |{y} | || x2 | | | +--------------+------+ || x3 | |{y, x}| || x2 | | | +--------------+------+ || -x | |{y} | || xy-y2+x | | | +--------------+------+ || x2 | |{y} | || y3 | | | +--------------+------+ || -x2 ||{y} | || x2y-xy2+x2 || | +--------------+------+ || 0 | |{y, x}| || x3+2x2+y2 | | | +--------------+------+ o9 : InvolutiveBasis i10 : F = factorModuleBasis J +------+--+ o10 = || 1 | |{}| || 0 | | | +------+--+ || y | |{}| || 0 | | | +------+--+ || y2 ||{}| || 0 || | +------+--+ || x | |{}| || 0 | | | +------+--+ || x2 ||{}| || 0 || | +------+--+ || 0 | |{}| || 1 | | | +------+--+ || 0 | |{}| || y | | | +------+--+ || 0 ||{}| || y2 || | +------+--+ || 0 | |{}| || x | | | +------+--+ || 0 ||{}| || x2 || | +------+--+ o10 : FactorModuleBasis i11 : basisElements F o11 = | 1 y y2 x x2 0 0 0 0 0 | | 0 0 0 0 0 1 y y2 x x2 | 2 10 o11 : Matrix R <-- R i12 : multVar F o12 = {set {}, set {}, set {}, set {}, set {}, set {}, set {}, set {}, set ----------------------------------------------------------------------- {}, set {}} o12 : List