$SPAD/src/input paff.input

Gaetan Hache December 3, 2016

Abstract

Contents

2

— * —

)set break resume
)spool paff.output
)set message test on
)set message auto off
)clear all
--S 1 of 26
K:= PACRAT
--R
--R
--R   (1)  PseudoAlgebraicClosureOfRationalNumber
--R                                                                 Type: Domain
--E 1
--S 2 of 26
symb:=[x,y,z]
--R
--R
--R   (2)  [x,y,z]
--R                                     Type: List(OrderedVariableList([x,y,z]))
--E 2
--S 3 of 26
PolyRing:=DMP(symb,K)
--R
--R
--R   (3)
--R  DistributedMultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNum
--R  ber)
--R                                                                 Type: Domain
--E 3
--S 4 of 26
E:=DIRPROD(#symb,NNI)
--R
--R
--R   (4)  DirectProduct(3,NonNegativeInteger)
--R                                                                 Type: Domain
--E 4
--S 5 of 26
ProjPt := ProjectivePlane(K)
--R
--R
--R   (5)  ProjectivePlane(PseudoAlgebraicClosureOfRationalNumber)
--R                                                                 Type: Domain
--E 5

--S 6 of 26
AFP:= AffinePlane(K)
--R
--R
--R   (6)  AffinePlane(PseudoAlgebraicClosureOfRationalNumber)
--R                                                                 Type: Domain
--E 6

———-

There are two possible BlowUp methods. The alternative method is invoked by:

BLMET :=  BLHN -- BlowUpWithHamburgerNoether

— * —

--S 7 of 26
BLMET :=  BLQT
--R
--R
--R   (7)  BlowUpWithQuadTrans
--R                                                                 Type: Domain
--E 7
--S 8 of 26
PCS := NeitherSparseOrDensePowerSeries(K)
--R
--R
--R   (8)
--R   NeitherSparseOrDensePowerSeries(PseudoAlgebraicClosureOfRationalNumber)
--R                                                                 Type: Domain
--E 8
--S 9 of 26
Plc:= PLACES K
--R
--R
--R   (9)  Places(PseudoAlgebraicClosureOfRationalNumber)
--R                                                                 Type: Domain
--E 9
--S 10 of 26
DIVISOR:= DIV Plc
--R
--R
--R   (10)  Divisor(Places(PseudoAlgebraicClosureOfRationalNumber))
--R                                                                 Type: Domain
--E 10

--S 11 of 26
InfClsPoint := InfClsPt(K,symb,BLMET)
--R
--R
--R   (11)
--R   InfClsPt(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],BlowUpWithQuadTrans)
--R                                                                 Type: Domain
--E 11
--S 12 of 26
DesTree := DesingTree InfClsPoint
--R
--R
--R   (12)
--R  DesingTree(InfClsPt(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],BlowUpWith
--R  QuadTrans))
--R                                                                 Type: Domain
--E 12
--S 13 of 26
BB:=BlowUpPackage(K,symb,PolyRing,E,BLMET)
--R
--R
--R   (13)
--R  BlowUpPackage(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],DistributedMulti
--R  variatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNumber),DirectProdu
--R  ct(3,NonNegativeInteger),BlowUpWithQuadTrans)
--R                                                                 Type: Domain
--E 13
--S 14 of 26
P:= GeneralPackageForAlgebraicFunctionField(K,symb,PolyRing,E,ProjPt, _
       PCS,Plc,DIVISOR,InfClsPoint,DesTree,BLMET)
--R
--R
--R   (14)
--R  GeneralPackageForAlgebraicFunctionField(PseudoAlgebraicClosureOfRationalNumbe
--R  r,[x,y,z],DistributedMultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOfR
--R  ationalNumber),DirectProduct(3,NonNegativeInteger),ProjectivePlane(PseudoAlge
--R  braicClosureOfRationalNumber),NeitherSparseOrDensePowerSeries(PseudoAlgebraic
--R  ClosureOfRationalNumber),Places(PseudoAlgebraicClosureOfRationalNumber),Divis
--R  or(Places(PseudoAlgebraicClosureOfRationalNumber)),InfClsPt(PseudoAlgebraicCl
--R  osureOfRationalNumber,[x,y,z],BlowUpWithQuadTrans),DesingTree(InfClsPt(Pseudo
--R  AlgebraicClosureOfRationalNumber,[x,y,z],BlowUpWithQuadTrans)),BlowUpWithQuad
--R  Trans)
--R                                                                 Type: Domain
--E 14
--S 15 of 26

ParamPackFC := _
  LocalParametrizationOfSimplePointPackage(K,symb,PolyRing,E,ProjPt,PCS,Plc)
--R
--R
--R   (15)
--R  LocalParametrizationOfSimplePointPackage(PseudoAlgebraicClosureOfRationalNumb
--R  er,[x,y,z],DistributedMultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOf
--R  RationalNumber),DirectProduct(3,NonNegativeInteger),ProjectivePlane(PseudoAlg
--R  ebraicClosureOfRationalNumber),NeitherSparseOrDensePowerSeries(PseudoAlgebrai
--R  cClosureOfRationalNumber),Places(PseudoAlgebraicClosureOfRationalNumber))
--R                                                                 Type: Domain
--E 15
--S 16 of 26
ParamPack := ParametrizationPackage(K,symb,PolyRing,E,ProjPt,PCS,Plc)
--R
--R
--R   (16)
--R  ParametrizationPackage(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],Distrib
--R  utedMultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNumber),Di
--R  rectProduct(3,NonNegativeInteger),ProjectivePlane(PseudoAlgebraicClosureOfRat
--R  ionalNumber),NeitherSparseOrDensePowerSeries(PseudoAlgebraicClosureOfRational
--R  Number),Places(PseudoAlgebraicClosureOfRationalNumber))
--R                                                                 Type: Domain
--E 16
--S 17 of 26
RatSingPack := ProjectiveAlgebraicSetPackage(K,symb,PolyRing,E,ProjPt)
--R
--R
--R   (17)
--R  ProjectiveAlgebraicSetPackage(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],
--R  DistributedMultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNum
--R  ber),DirectProduct(3,NonNegativeInteger),ProjectivePlane(PseudoAlgebraicClosu
--R  reOfRationalNumber))
--R                                                                 Type: Domain
--E 17
--S 18 of 26
IntDivPack := IntersectionDivisorPackage(K,symb,PolyRing,E,ProjPt,PCS,_
                                         Plc,DIVISOR,InfClsPoint,DesTree,BLMET)
--R
--R
--R   (18)
--R  IntersectionDivisorPackage(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],Dis
--R  tributedMultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNumber
--R  ),DirectProduct(3,NonNegativeInteger),ProjectivePlane(PseudoAlgebraicClosureO
--R  fRationalNumber),NeitherSparseOrDensePowerSeries(PseudoAlgebraicClosureOfRati
--R  onalNumber),Places(PseudoAlgebraicClosureOfRationalNumber),Divisor(Places(Pse
--R  udoAlgebraicClosureOfRationalNumber)),InfClsPt(PseudoAlgebraicClosureOfRation

--R  alNumber,[x,y,z],BlowUpWithQuadTrans),DesingTree(InfClsPt(PseudoAlgebraicClos
--R  ureOfRationalNumber,[x,y,z],BlowUpWithQuadTrans)),BlowUpWithQuadTrans)
--R                                                                 Type: Domain
--E 18
--S 19 of 26
IntFrmPack := InterpolateFormsPackage(K,symb,PolyRing,E,ProjPt,PCS,Plc,DIVISOR)
--R
--R
--R   (19)
--R  InterpolateFormsPackage(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],Distri
--R  butedMultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNumber),D
--R  irectProduct(3,NonNegativeInteger),ProjectivePlane(PseudoAlgebraicClosureOfRa
--R  tionalNumber),NeitherSparseOrDensePowerSeries(PseudoAlgebraicClosureOfRationa
--R  lNumber),Places(PseudoAlgebraicClosureOfRationalNumber),Divisor(Places(Pseudo
--R  AlgebraicClosureOfRationalNumber)))
--R                                                                 Type: Domain
--E 19
--S 20 of 26
DesTrPack:= DesingTreePackage(K,symb,PolyRing,E,ProjPt,PCS,Plc,DIVISOR,_
                              InfClsPoint,DesTree,BLMET)
--R
--R
--R   (20)
--R  DesingTreePackage(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],DistributedM
--R  ultivariatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNumber),DirectP
--R  roduct(3,NonNegativeInteger),ProjectivePlane(PseudoAlgebraicClosureOfRational
--R  Number),NeitherSparseOrDensePowerSeries(PseudoAlgebraicClosureOfRationalNumbe
--R  r),Places(PseudoAlgebraicClosureOfRationalNumber),Divisor(Places(PseudoAlgebr
--R  aicClosureOfRationalNumber)),InfClsPt(PseudoAlgebraicClosureOfRationalNumber,
--R  [x,y,z],BlowUpWithQuadTrans),DesingTree(InfClsPt(PseudoAlgebraicClosureOfRati
--R  onalNumber,[x,y,z],BlowUpWithQuadTrans)),BlowUpWithQuadTrans)
--R                                                                 Type: Domain
--E 20
--S 21 of 26
PackPoly    := PackageForPoly(K,PolyRing,E,#symb)
--R
--R
--R   (21)
--R  PackageForPoly(PseudoAlgebraicClosureOfRationalNumber,DistributedMultivariate
--R  Polynomial([x,y,z],PseudoAlgebraicClosureOfRationalNumber),DirectProduct(3,No
--R  nNegativeInteger),3)
--R                                                                 Type: Domain
--E 21
--S 22 of 26
BB:= BlowUpPackage(K,symb,PolyRing,E, BLMET)
--R

--R
--R   (22)
--R  BlowUpPackage(PseudoAlgebraicClosureOfRationalNumber,[x,y,z],DistributedMulti
--R  variatePolynomial([x,y,z],PseudoAlgebraicClosureOfRationalNumber),DirectProdu
--R  ct(3,NonNegativeInteger),BlowUpWithQuadTrans)
--R                                                                 Type: Domain
--E 22
--S 23 of 26
LinSer      := LinearSystemFromPowerSeriesPackage(K,PCS)
--R
--R
--R   (23)
--R  LinearSystemFromPowerSeriesPackage(PseudoAlgebraicClosureOfRationalNumber,Nei
--R  therSparseOrDensePowerSeries(PseudoAlgebraicClosureOfRationalNumber))
--R                                                                 Type: Domain
--E 23
--S 24 of 26
f:PolyRing:= x**28*z**8 + 4*x**26*z**10 + 6*x**24*z**12 + 4*x**22*z**14 + _
             4*x**21*y**9*z**6 + x**20*z**16 + 12*x**19*y**9*z**8 + _
             12*x**17*y**9*z**10 + 4*x**15*y**9*z**12 + 6*x**14*y**18*z**4 + _
             12*x**12*y**18*z**6 + 6*x**10*y**18*z**8 + 4*x**7*y**27*z*z + _
             4*x**5*y**27*z**4 + y**36 + (-2*y**20*z**16)
--R
--R
--R   (24)
--R      28 8     26 10     24 12     22 14     21 9 6    20 16      19 9 8
--R     x  z  + 4x  z   + 6x  z   + 4x  z   + 4x  y z  + x  z   + 12x  y z
--R   +
--R        17 9 10     15 9 12     14 18 4      12 18 6     10 18 8     7 27 2
--R     12x  y z   + 4x  y z   + 6x  y  z  + 12x  y  z  + 6x  y  z  + 4x y  z
--R   +
--R       5 27 4    36     20 16
--R     4x y  z  + y   - 2y  z
--IType: DistributedMultivariatePolynomial(...
--E 24
--S 25 of 26
fh:=homogenize(f,3)$P
--R
--R
--R   (25)
--R      28 8     26 10     24 12     22 14     21 9 6    20 16      19 9 8
--R     x  z  + 4x  z   + 6x  z   + 4x  z   + 4x  y z  + x  z   + 12x  y z
--R   +
--R        17 9 10     15 9 12     14 18 4      12 18 6     10 18 8     7 27 2
--R     12x  y z   + 4x  y z   + 6x  y  z  + 12x  y  z  + 6x  y  z  + 4x y  z
--R   +
--R       5 27 4    36     20 16

--R     4x y  z  + y   - 2y  z
--IType: DistributedMultivariatePolynomial([x,y,z],...
--E 25
--S 26 of 26
setCurve(fh)$P
--R
--R
--R   (26)
--R      28 8     26 10     24 12     22 14     21 9 6    20 16      19 9 8
--R     x  z  + 4x  z   + 6x  z   + 4x  z   + 4x  y z  + x  z   + 12x  y z
--R   +
--R        17 9 10     15 9 12     14 18 4      12 18 6     10 18 8     7 27 2
--R     12x  y z   + 4x  y z   + 6x  y  z  + 12x  y  z  + 6x  y  z  + 4x y  z
--R   +
--R       5 27 4    36     20 16
--R     4x y  z  + y   - 2y  z
--IType: DistributedMultivariatePolynomial([x,y,z],...
--E 26
)spool
)lisp (bye)

———-

References^{[1] nothing}

10