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RationalMaps :: jacobianDualMatrix

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

Synopsis

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 :

For the programmer

The object jacobianDualMatrix is a method function with options.