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RegularChains[ChainTools]

  

DahanSchostTransform

  

map a regular chain by the Dahan and Schost transform

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

DahanSchostTransform(rc, R)

Parameters

rc

-

regular chain of R

R

-

polynomial ring

Description

• 

The command DahanSchostTransform(rc, R) returns the regular chain obtained by applying the Dahan and Schost transform to rc.

• 

The output regular chain has the same saturated ideal as the input rc. Moreover, the size of the coefficients of the output regular chain is very likely to be much smaller than that of rc.

• 

This function assumes that rc is zero-dimensional and normalized, and that the saturated ideal of rc is radical.

• 

This command is part of the RegularChains[ChainTools] package, so it can be used in the form DahanSchostTransform(..) only after executing the command with(RegularChains[ChainTools]). However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][DahanSchostTransform](..).

Examples

withRegularChains:withChainTools:

RPolynomialRingx,y,z

R:=polynomial_ring

(1)

sysx5+y53y1,5y43,20x+yz

sys:=20x+yz,5y43,x5+y53y1

(2)

decTriangularizesys,R,normalized=yes

dec:=regular_chain

(3)

mapEquations,dec,R

11474127946569256007468861967138822599454632253404776870051199476222619269004890144761853439484671057123097693465191381050813704562732917125293370932479130541598163953960078201654747916507320573574680356823040x1771260505008202862102854051702189834144507041921400912212854357946960933195335641858396501896935850288388698973423657024874890168951887006119520716403898588447265625000000000000z19+6993494167255643877060419555161219397297718310661681373013610473433161675295215097739765468198629739368658992144459186504824936653303907728642932110182220458984375000000000000000z18+46980330573720043696285723094038459435169014560960809457932826698816864853909365786661752359672134274602580990491031583381442373098978089863389116690573928381886816436810143422403125z17362457794998087226523064237197118238681455387434685379217170814307753153223785029557758914206492139656047158750881706208736371102082285270209948560334845112137055928302241429687500000z16+1825588409831441292570286016853843732976447711290921201282663597873225040956392206905741146687704996955957543164892852708765201681735455663532400560561990184217739346093750000000000000z15+15138417846066725118358222658899878896246722526651227781338839693046020627409354976198946514427454581361714047528896855158315263311055631123529796231583257599303710937500000000000000000z1444394335873903477558622382037619903399605543513019193984850811034401539767435244582975861827087564468519794770271434211676417079225209120663140736918184704321681411620260946061817664157500z13239889463831973885970439654459159240773157947028995584430781544269432684180568707791767576191787113033986469119189293532827387996841920408461551066623181822838810854143891280165251250000000z12+2738339662798997128827712967353520807578712156161195412624338459316853569080754130154719452119622862823530609072130944805722667158059695598771740436061650119196713974446586975000000000000000z1115237133948658997778693395344596342126523231688102858941028295140149607477956051848066457333497202284356641427141996353739796250479018624174619989841534614074349477755996093750000000000000000z10+48563913474106327770615609511108962756349408870293446119857242983280899281287041276597414703953142847110970543541965415369819771789164016371916892583484885932065222266474244223656870580763404750z9+182770901475269211462030828375934181004032581754339209581456763239413822566355167569080400536438012882499416890307268474234594753941991910223913471336544588355841793697886482347774623132375000000z8+309191296130950729973668595368021125635249693248658751381279239017170403224531631090451630403456902301090463541541208661015512101696917723097258264871336817986092045758275049045390250000000000000z76838688396641645490945090868618366582490420637673970853279869471018348887091817749546675847593376908651767451450243404728169565283801174082247345442363224813502338696761842209062500000000000000000z674815682380070752593065205631091355818115420146560706379886171073303776505335730603765529125626467971633284964641482528987570423158923499017031906462266550721041958679333491247172414080955401845651420z5154608045527569292338754337973797843824713701855230758768236174292780150592090630056630234512064066763987236066826571445077192641952424283276680662979283353386614928379571789390957015570270631450000000z4124695385819578642285275287975402015668994502200477065094640515598601115130175167063705343665239193213631330215705606292594773399242228508627801381881595789007869020418417852791444860110175000000000000z366152659857188245320424888024222967738184293737891699176976594293187674688484864881423871033576765065422479234984449961643090211269389726039235842212801812250522818211946896210922615625000000000000000z2+573598714920124956474610718803150703376812978417179178775576117319500000077857129232958889104193427114987500929833686714791341712743162700766075396541379832681132358445310329142895528887477470724804102079717z+239787108649287987286424755607482454864690786827841184696976286133386057573817722098997859322480446751288360706756986017238407677469601203785469989972240351121714956603188344506951151541954650850823956500000,573706397328462800373443098356941129972731612670238843502559973811130963450244507238092671974233552856154884673259569052540685228136645856264668546623956527079908197698003910082737395825366028678734017841152y1771260505008202862102854051702189834144507041921400912212854357946960933195335641858396501896935850288388698973423657024874890168951887006119520716403898588447265625000000000000z19+6993494167255643877060419555161219397297718310661681373013610473433161675295215097739765468198629739368658992144459186504824936653303907728642932110182220458984375000000000000000z18+46980330573720043696285723094038459435169014560960809457932826698816864853909365786661752359672134274602580990491031583381442373098978089863389116690573928381886816436810143422403125z1736245779499808722652306423719711823868145538743468537921717081430775315322378502955775891420649213965604715875088170620873637110208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(4)

dsDahanSchostTransformdec1,R

ds:=regular_chain

(5)

Equationsds,R

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,3125z197500z1530000000000z141209599993250z11858000000000z10+96000000000000000z94866050142720002700z75160960781200000000z61958400000000000000z5102400000000000000000000z43397381727846669567999595z32123365072896045360000000z2589823873280000000000000z61440000000000000000000y1875z16302399995500z12312000000000z111216513874880004050z82211840892800000000z71305600000000000000z6849339791770341311998380z41415574503424181440000000z3884735493120000000000000z2245760000000000000000000z509608055439369331200243,3125z209375z1640000000000z152015999988750z121560000000000z11+192000000000000000z1012165125356800006750z814745602232000000000z76528000000000000000z6409600000000000000000000z516986908639233347839997975z414155767152640302400000000z35898238732800000000000000z21228800000000000000000000z6195303619231982878732441600243

(6)

EqualSaturatedIdealsdec1,ds,R

true

(7)

References

  

Dahan, X., and Schost, E. "Sharp Estimates for Triangular Sets." In Proc. ISSAC 2004, Santander, Spain, ACM Press, 2004.

See Also

ChainTools

EqualSaturatedIdeals

Equations

IsStronglyNormalized

IsZeroDimensional

PolynomialRing

RegularChains

Triangularize

 


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