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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left\{5y^4-3\,-20x+y-z\,-x^5+y^5-3y-1\right\}$

$\mathrm{sys}≔\left\{-20x+y-z\,5y^4-3\,-x^5+y^5-3y-1\right\}$

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{normalized}=\mathrm{yes}\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{dec}\,R\right)$

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$\mathrm{ds}≔\mathrm{DahanSchostTransform}\left(\mathrm{dec}\left[1\right]\,R\right)$

$\mathrm{ds}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{ds}\,R\right)$

$\left[\left(625z^{19}-1500z^{15}-6000000000z^{14}-241919998650z^{11}-171600000000z^{10}+19200000000000000z^9-973210028544000540z^7-1032192156240000000z^6-391680000000000000z^5-20480000000000000000000z^4-679476345569333913599919z^3-424673014579209072000000z^2-117964774656000000000000z-12288000000000000000000\right)x+100000000z^{15}+5040000000z^{12}+3900000000z^{11}-960000000000000z^{10}+60825613392000000z^8+73728005580000000z^7+32640000000000000z^6+3072000000000000000000z^5+127401871196163369600000z^4+106168275763200756000000z^3+44236793664000000000000z^2+9216000000000000000000z+61953031096239274393631104000\,\left(3125z^{19}-7500z^{15}-30000000000z^{14}-1209599993250z^{11}-858000000000z^{10}+96000000000000000z^9-4866050142720002700z^7-5160960781200000000z^6-1958400000000000000z^5-102400000000000000000000z^4-3397381727846669567999595z^3-2123365072896045360000000z^2-589823873280000000000000z-61440000000000000000000\right)y-1875z^{16}-302399995500z^{12}-312000000000z^{11}-1216513874880004050z^8-2211840892800000000z^7-1305600000000000000z^6-849339791770341311998380z^4-1415574503424181440000000z^3-884735493120000000000000z^2-245760000000000000000000z-509608055439369331200243\,3125z^{20}-9375z^{16}-40000000000z^{15}-2015999988750z^{12}-1560000000000z^{11}+192000000000000000z^{10}-12165125356800006750z^8-14745602232000000000z^7-6528000000000000000z^6-409600000000000000000000z^5-16986908639233347839997975z^4-14155767152640302400000000z^3-5898238732800000000000000z^2-1228800000000000000000000z-6195303619231982878732441600243\right]$

$\mathrm{EqualSaturatedIdeals}\left(\mathrm{dec}\left[1\right]\,\mathrm{ds}\,R\right)$

$\mathrm{true}$

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