DahanSchostTransform - Maple Help

RegularChains[ChainTools]

 DahanSchostTransform
 map a regular chain by the Dahan and Schost transform

 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

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{sys}≔\left\{5{y}^{4}-3,-20x+y-z,-{x}^{5}+{y}^{5}-3y-1\right\}$
 ${\mathrm{sys}}{≔}\left\{{-}{20}{}{x}{+}{y}{-}{z}{,}{5}{}{{y}}^{{4}}{-}{3}{,}{-}{{x}}^{{5}}{+}{{y}}^{{5}}{-}{3}{}{y}{-}{1}\right\}$ (2)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{normalized}=\mathrm{yes}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 $\left[\left[{11474127946569256007468861967138822599454632253404776870051199476222619269004890144761853439484671057123097693465191381050813704562732917125293370932479130541598163953960078201654747916507320573574680356823040}{}{x}{-}{1771260505008202862102854051702189834144507041921400912212854357946960933195335641858396501896935850288388698973423657024874890168951887006119520716403898588447265625000000000000}{}{{z}}^{{19}}{+}{6993494167255643877060419555161219397297718310661681373013610473433161675295215097739765468198629739368658992144459186504824936653303907728642932110182220458984375000000000000000}{}{{z}}^{{18}}{+}{46980330573720043696285723094038459435169014560960809457932826698816864853909365786661752359672134274602580990491031583381442373098978089863389116690573928381886816436810143422403125}{}{{z}}^{{17}}{-}{362457794998087226523064237197118238681455387434685379217170814307753153223785029557758914206492139656047158750881706208736371102082285270209948560334845112137055928302241429687500000}{}{{z}}^{{16}}{+}{1825588409831441292570286016853843732976447711290921201282663597873225040956392206905741146687704996955957543164892852708765201681735455663532400560561990184217739346093750000000000000}{}{{z}}^{{15}}{+}{15138417846066725118358222658899878896246722526651227781338839693046020627409354976198946514427454581361714047528896855158315263311055631123529796231583257599303710937500000000000000000}{}{{z}}^{{14}}{-}{44394335873903477558622382037619903399605543513019193984850811034401539767435244582975861827087564468519794770271434211676417079225209120663140736918184704321681411620260946061817664157500}{}{{z}}^{{13}}{-}{239889463831973885970439654459159240773157947028995584430781544269432684180568707791767576191787113033986469119189293532827387996841920408461551066623181822838810854143891280165251250000000}{}{{z}}^{{12}}{+}{2738339662798997128827712967353520807578712156161195412624338459316853569080754130154719452119622862823530609072130944805722667158059695598771740436061650119196713974446586975000000000000000}{}{{z}}^{{11}}{-}{15237133948658997778693395344596342126523231688102858941028295140149607477956051848066457333497202284356641427141996353739796250479018624174619989841534614074349477755996093750000000000000000}{}{{z}}^{{10}}{+}{48563913474106327770615609511108962756349408870293446119857242983280899281287041276597414703953142847110970543541965415369819771789164016371916892583484885932065222266474244223656870580763404750}{}{{z}}^{{9}}{+}{182770901475269211462030828375934181004032581754339209581456763239413822566355167569080400536438012882499416890307268474234594753941991910223913471336544588355841793697886482347774623132375000000}{}{{z}}^{{8}}{+}{309191296130950729973668595368021125635249693248658751381279239017170403224531631090451630403456902301090463541541208661015512101696917723097258264871336817986092045758275049045390250000000000000}{}{{z}}^{{7}}{-}{6838688396641645490945090868618366582490420637673970853279869471018348887091817749546675847593376908651767451450243404728169565283801174082247345442363224813502338696761842209062500000000000000000}{}{{z}}^{{6}}{-}{74815682380070752593065205631091355818115420146560706379886171073303776505335730603765529125626467971633284964641482528987570423158923499017031906462266550721041958679333491247172414080955401845651420}{}{{z}}^{{5}}{-}{154608045527569292338754337973797843824713701855230758768236174292780150592090630056630234512064066763987236066826571445077192641952424283276680662979283353386614928379571789390957015570270631450000000}{}{{z}}^{{4}}{-}{124695385819578642285275287975402015668994502200477065094640515598601115130175167063705343665239193213631330215705606292594773399242228508627801381881595789007869020418417852791444860110175000000000000}{}{{z}}^{{3}}{-}{66152659857188245320424888024222967738184293737891699176976594293187674688484864881423871033576765065422479234984449961643090211269389726039235842212801812250522818211946896210922615625000000000000000}{}{{z}}^{{2}}{+}{573598714920124956474610718803150703376812978417179178775576117319500000077857129232958889104193427114987500929833686714791341712743162700766075396541379832681132358445310329142895528887477470724804102079717}{}{z}{+}{239787108649287987286424755607482454864690786827841184696976286133386057573817722098997859322480446751288360706756986017238407677469601203785469989972240351121714956603188344506951151541954650850823956500000}{,}{573706397328462800373443098356941129972731612670238843502559973811130963450244507238092671974233552856154884673259569052540685228136645856264668546623956527079908197698003910082737395825366028678734017841152}{}{y}{-}{1771260505008202862102854051702189834144507041921400912212854357946960933195335641858396501896935850288388698973423657024874890168951887006119520716403898588447265625000000000000}{}{{z}}^{{19}}{+}{699349416725564387706041955516121939729771831066168137301361047343316167529521509773976546819862973936865899214445918650482493665330390772864293211018222045898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(4)
 > $\mathrm{ds}≔\mathrm{DahanSchostTransform}\left(\mathrm{dec}\left[1\right],R\right)$
 ${\mathrm{ds}}{≔}{\mathrm{regular_chain}}$ (5)
 > $\mathrm{Equations}\left(\mathrm{ds},R\right)$
 $\left[\left({625}{}{{z}}^{{19}}{-}{1500}{}{{z}}^{{15}}{-}{6000000000}{}{{z}}^{{14}}{-}{241919998650}{}{{z}}^{{11}}{-}{171600000000}{}{{z}}^{{10}}{+}{19200000000000000}{}{{z}}^{{9}}{-}{973210028544000540}{}{{z}}^{{7}}{-}{1032192156240000000}{}{{z}}^{{6}}{-}{391680000000000000}{}{{z}}^{{5}}{-}{20480000000000000000000}{}{{z}}^{{4}}{-}{679476345569333913599919}{}{{z}}^{{3}}{-}{424673014579209072000000}{}{{z}}^{{2}}{-}{117964774656000000000000}{}{z}{-}{12288000000000000000000}\right){}{x}{+}{100000000}{}{{z}}^{{15}}{+}{5040000000}{}{{z}}^{{12}}{+}{3900000000}{}{{z}}^{{11}}{-}{960000000000000}{}{{z}}^{{10}}{+}{60825613392000000}{}{{z}}^{{8}}{+}{73728005580000000}{}{{z}}^{{7}}{+}{32640000000000000}{}{{z}}^{{6}}{+}{3072000000000000000000}{}{{z}}^{{5}}{+}{127401871196163369600000}{}{{z}}^{{4}}{+}{106168275763200756000000}{}{{z}}^{{3}}{+}{44236793664000000000000}{}{{z}}^{{2}}{+}{9216000000000000000000}{}{z}{+}{61953031096239274393631104000}{,}\left({3125}{}{{z}}^{{19}}{-}{7500}{}{{z}}^{{15}}{-}{30000000000}{}{{z}}^{{14}}{-}{1209599993250}{}{{z}}^{{11}}{-}{858000000000}{}{{z}}^{{10}}{+}{96000000000000000}{}{{z}}^{{9}}{-}{4866050142720002700}{}{{z}}^{{7}}{-}{5160960781200000000}{}{{z}}^{{6}}{-}{1958400000000000000}{}{{z}}^{{5}}{-}{102400000000000000000000}{}{{z}}^{{4}}{-}{3397381727846669567999595}{}{{z}}^{{3}}{-}{2123365072896045360000000}{}{{z}}^{{2}}{-}{589823873280000000000000}{}{z}{-}{61440000000000000000000}\right){}{y}{-}{1875}{}{{z}}^{{16}}{-}{302399995500}{}{{z}}^{{12}}{-}{312000000000}{}{{z}}^{{11}}{-}{1216513874880004050}{}{{z}}^{{8}}{-}{2211840892800000000}{}{{z}}^{{7}}{-}{1305600000000000000}{}{{z}}^{{6}}{-}{849339791770341311998380}{}{{z}}^{{4}}{-}{1415574503424181440000000}{}{{z}}^{{3}}{-}{884735493120000000000000}{}{{z}}^{{2}}{-}{245760000000000000000000}{}{z}{-}{509608055439369331200243}{,}{3125}{}{{z}}^{{20}}{-}{9375}{}{{z}}^{{16}}{-}{40000000000}{}{{z}}^{{15}}{-}{2015999988750}{}{{z}}^{{12}}{-}{1560000000000}{}{{z}}^{{11}}{+}{192000000000000000}{}{{z}}^{{10}}{-}{12165125356800006750}{}{{z}}^{{8}}{-}{14745602232000000000}{}{{z}}^{{7}}{-}{6528000000000000000}{}{{z}}^{{6}}{-}{409600000000000000000000}{}{{z}}^{{5}}{-}{16986908639233347839997975}{}{{z}}^{{4}}{-}{14155767152640302400000000}{}{{z}}^{{3}}{-}{5898238732800000000000000}{}{{z}}^{{2}}{-}{1228800000000000000000000}{}{z}{-}{6195303619231982878732441600243}\right]$ (6)
 > $\mathrm{EqualSaturatedIdeals}\left(\mathrm{dec}\left[1\right],\mathrm{ds},R\right)$
 ${\mathrm{true}}$ (7)

References

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