# jacobianDualMatrix -- Computes the Jacobian Dual Matrix, a matrix whose kernel describing the syzygies of the inverse map.

## Synopsis

• Usage:
M = jacobianDualMatrix(a,b,g)
M = jacobianDualMatrix(R,S,g)
M = jacobianDualMatrix(p)
• Inputs:
• a, an ideal, defining equations for X
• b, an ideal, defining equations for Y
• g, , projective rational map given by polynomial representatives
• R, a ring, coordinate ring of X
• S, a ring, coordinate ring of Y
• p, , projective rational map given by polynomial representatives
• Optional inputs:
• AssumeDominant => ..., default value false, If true, certain functions assume that the map from X to Y is dominant.
• QuickRank => ..., default value true, An option for computing how rank is computed
• Strategy => ..., default value ReesStrategy, Determines the desired Strategy in each function.
• Outputs:
• M, , Returns a matrix over the coordinate ring of the image, the kernel of this matrix describing the syzygies of the inverse map, if it exists.

## Description

This is mostly an internal function which is used when checking if a map is birational and when computing the inverse map. If the AssumeDominant option is set to true, it assumes that the kernel of the associated ring map is zero (default value is false). Valid values for the Strategy option are ReesStrategy and SaturationStrategy. For more information, see Doria, Hassanzadeh, Simis, A characteristic-free criterion of birationality. Adv. Math. 230 (2012), no. 1, 390–413.

 i1 : R=QQ[x,y]; i2 : S=QQ[a,b,c,d]; i3 : Pi = map(R, S, {x^3, x^2*y, x*y^2, y^3}); o3 : RingMap R <--- S i4 : jacobianDualMatrix(Pi, Strategy=>SaturationStrategy) o4 = | -d -c -b | | c b a | / S \2 / S \3 o4 : Matrix |-------------------------------| <--- |-------------------------------| | 2 2 | | 2 2 | \(c - b*d, b*c - a*d, b - a*c)/ \(c - b*d, b*c - a*d, b - a*c)/

## Ways to use jacobianDualMatrix :

• "jacobianDualMatrix(Ideal,Ideal,BasicList)"
• "jacobianDualMatrix(Ring,Ring,BasicList)"
• "jacobianDualMatrix(RingMap)"

## For the programmer

The object jacobianDualMatrix is .