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# jacobianDualMatrix -- computes the Jacobian dual matrix

## Synopsis

• Usage:
M = jacobianDualMatrix(p)
M = jacobianDualMatrix(phi)
• Inputs:
• phi, an instance of the type RationalMapping, a rational map between projective varieties
• p, , ring map corresponding to a rational map between projective varieties
• Optional inputs:
• Strategy => , default value ReesStrategy, choose the strategy to use: ReesStrategy or SaturationStrategy
• AssumeDominant => , default value false, whether to assume the provided rational map of projective varieties is dominant, if set to true it can speed up computation
• QuickRank => , default value true, whether to compute rank via the package FastMinors
• Outputs:
• M, , a matrix $M$ over the coordinate ring of the image, the kernel of $M$ describes the syzygies of the inverse map, if it exists.

## Description

The Jacobian dual matrix is a matrix whose kernel describes the syzygies of the matrix corresponding to the inverse map. For more information, see

• Doria, A. V.; Hassanzadeh, S. H.; Simis, A. A characteristic-free criterion of birationality. Adv. Math. 230 (2012), no. 1, 390--413.

This is mostly an internal function. It is used when checking whether 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.

 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 0 0 0 | | c b a 0 0 0 | / S \2 / S \6 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)/