inverse of a polynomial with respect to a regular chain
Inverse(p, rc, R)
Inverse(p, rc, R, 'normalized'='yes')
regular chain of R
polynomial of R
boolean flag (optional)
The function call Inverse(p, rc, R) returns a list inv,zdiv. The list inv consists of pairs qi,hi,rci such that qi⁢p equals hi modulo the saturated ideal of rci, where hi is regular with respect to rci. The list zdiv is a list of regular chains rcj such that p is a zero-divisor modulo rcj. In addition, the set of all regular chains occurring in inv and zdiv is a triangular decomposition of rc. To be precise, they form a decomposition of rc in the sense of Kalkbrener.
If 'normalized'='yes' is passed, then the regular chain rc must be normalized. In addition, all the returned regular chains will be normalized.
If the regular chain rc is normalized but 'normalized'='yes' is not passed, then there is no guarantee that the returned regular chains will be normalized.
For zero-dimensional regular chains in prime characteristic, the commands RegularizeDim0 and NormalizePolynomialDim0 can be combined to obtain the same specification as the command Inverse while gaining the advantages of modular techniques and asymptotically fast polynomial arithmetic.
This command is part of the RegularChains package, so it can be used in the form Inverse(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[Inverse](..).
R ≔ PolynomialRing⁡x,y,z
R ≔ polynomial_ring
rc ≔ Chain⁡z2+1,y2+z,Empty⁡R,R
rc ≔ regular_chain
p ≔ y−z
rc ≔ Chain⁡z2+1,y2+1,Empty⁡R,R
inv,zdiv ≔ op⁡Inverse⁡p,rc,R:
q1 ≔ inv11;h1 ≔ inv12;rc1 ≔ inv13;Equations⁡rc1,R
q1 ≔ z
h1 ≔ 2
rc1 ≔ regular_chain
rc2 ≔ zdiv1;Equations⁡rc2,R
rc2 ≔ regular_chain
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