Groebner - Maple Help

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Groebner

 Walk
 convert Groebner bases from one ordering to another

 Calling Sequence Walk(G, T1, T2, opts)

Parameters

 G - Groebner basis with respect to starting order T1 or a PolynomialIdeal T1,T2 - monomial orders (of type ShortMonomialOrder) opts - optional arguments of the form keyword=value

Description

 • The Groebner walk algorithm converts a Groebner basis of commutative polynomials from one monomial order to another.  It is frequently applied when a Groebner basis is too difficult to compute directly.
 • The Walk command takes as input a Groebner basis G with respect to a monomial order T1, and outputs the reduced Groebner basis for G with respect to T2.  If the first argument G is a PolynomialIdeal then a Groebner basis for G with respect to T1 is computed if one is not already known.
 • The orders T1 and T2 must be proper monomial orders on the polynomial ring, so 'min' orders such as 'plex_min' and 'tdeg_min' are not supported. Walk does not check that G is a Groebner basis with respect to T1.
 • Unlike FGLM, the ideal defined by G can have an infinite number of solutions. The Groebner walk is typically not as fast as FGLM on zero-dimensional ideals.
 • The optional argument characteristic=p specifies the characteristic of the coefficient field. The default is zero.  This option is ignored if G is a PolynomialIdeal.
 • The optional argument elimination=true forces the Groebner walk to terminate early, before a Groebner basis with respect to T2 is obtained.  If T2 is a lexdeg order with two blocks of variables the resulting list will contain a generating set of the elimination ideal.
 • The optional argument output=basislm returns the basis in an extended format containing leading monomials and coefficients.  Each element is a list of the form [leading coefficient, leading monomial, polynomial].
 • Setting infolevel[Walk] to a positive integer value directs the Walk command to output increasingly detailed information about its performance and progress.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $\mathrm{F1}≔\left[10xz-6{x}^{3}-8{y}^{2}{z}^{2},-6z+5{y}^{3}\right]$
 ${\mathrm{F1}}{:=}\left[{-}{8}{}{{y}}^{{2}}{}{{z}}^{{2}}{-}{6}{}{{x}}^{{3}}{+}{10}{}{x}{}{z}{,}{5}{}{{y}}^{{3}}{-}{6}{}{z}\right]$ (1)
 > $\mathrm{G1}≔\mathrm{Basis}\left(\mathrm{F1},\mathrm{tdeg}\left(x,y,z\right)\right)$
 ${\mathrm{G1}}{:=}\left[{5}{}{{y}}^{{3}}{-}{6}{}{z}{,}{4}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{3}{}{{x}}^{{3}}{-}{5}{}{x}{}{z}{,}{15}{}{{x}}^{{3}}{}{y}{-}{25}{}{x}{}{y}{}{z}{+}{24}{}{{z}}^{{3}}{,}{45}{}{{x}}^{{6}}{-}{96}{}{y}{}{{z}}^{{5}}{-}{150}{}{{x}}^{{4}}{}{z}{+}{125}{}{{x}}^{{2}}{}{{z}}^{{2}}\right]$ (2)
 > $\mathrm{Walk}\left(\mathrm{G1},\mathrm{tdeg}\left(x,y,z\right),\mathrm{plex}\left(x,y,z\right)\right)$
 $\left[{5}{}{{y}}^{{3}}{-}{6}{}{z}{,}{4}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{3}{}{{x}}^{{3}}{-}{5}{}{x}{}{z}\right]$ (3)
 > $\mathrm{alias}\left(\mathrm{α}=\mathrm{RootOf}\left({z}^{2}+z+5\right)\right)$
 ${\mathrm{α}}$ (4)
 > $\mathrm{F2}≔\left[-10yx-9{x}^{3}+2z{\mathrm{α}}^{2}-4{y}^{3}\mathrm{α},6{\mathrm{α}}^{2}-2{x}^{2}\mathrm{α}+9x{\mathrm{α}}^{2}-8{y}^{3}x\right]:$
 > $\mathrm{G2}≔\mathrm{Basis}\left(\mathrm{F2},\mathrm{tdeg}\left(x,y,z\right)\right)$
 ${\mathrm{G2}}{:=}\left[\left({2}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{10}\right){}{z}{+}{9}{}{{x}}^{{3}}{+}{4}{}{{y}}^{{3}}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{10}{}{y}{}{x}{,}{6}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{30}{+}{2}{}{{x}}^{{2}}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}\left({9}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{45}\right){}{x}{+}{8}{}{{y}}^{{3}}{}{x}{,}{-}{24}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{30}{+}{32}{}{{y}}^{{6}}{+}\left({56}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{10}\right){}{{x}}^{{2}}{+}\left({-}{72}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{90}\right){}{z}{+}\left({-}{36}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{45}\right){}{x}{+}{60}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{-}{16}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{z}{+}\left({36}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{180}\right){}{{y}}^{{3}}{+}\left({4}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{20}\right){}{x}{}{z}\right]$ (5)
 > $\mathrm{Walk}\left(\mathrm{G2},\mathrm{tdeg}\left(x,y,z\right),\mathrm{plex}\left(x,y,z\right)\right)$
 $\left[{-}{393660}{-}{4608}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{}{z}{-}{62208}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{}{z}{+}{1280}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{}{z}{+}{3960}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{}{z}{+}{31104}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{+}{16640}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{7}}{+}{293616}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{+}{149040}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{-}{4800}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{2}}{+}{896}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{z}}^{{2}}{+}{9216}{}{{y}}^{{12}}{-}{1600}{}{{y}}^{{4}}{}{z}{-}{208800}{}{{y}}^{{4}}{-}{16200}{}{y}{}{z}{+}{77760}{}{{y}}^{{6}}{}{z}{+}{155520}{}{{y}}^{{9}}{+}{724480}{}{{y}}^{{6}}{+}{96228}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{880}{}{{z}}^{{2}}{+}{1676700}{}{z}{-}{854550}{}{y}{-}{156735}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{z}{+}{298890}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{+}{732600}{}{{y}}^{{3}}{}{z}{-}{215080}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{z}{-}{3200}{}{{y}}^{{7}}{+}{984150}{}{{y}}^{{3}}{+}{6000}{}{{y}}^{{2}}{+}{670680}{}{{y}}^{{3}}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){,}{5550200727030000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{{z}}^{{2}}{-}{11336483045680000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{2}}{}{z}{-}{1020432054600000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{}{{z}}^{{2}}{+}{1947158001561600}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{}{z}{+}{16968987770878080}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{}{z}{+}{45006197725728000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{}{z}{+}{95020410601170000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{}{z}{-}{314125516800}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{}{{z}}^{{4}}{-}{15975784120320}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{10}}{}{{z}}^{{2}}{-}{5536014336000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{}{{z}}^{{3}}{-}{32882551603200}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{10}}{}{z}{-}{10435297443840}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{}{{z}}^{{2}}{-}{322486272000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{}{{z}}^{{5}}{-}{4917019852800}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{7}}{}{{z}}^{{3}}{+}{1310100480000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{}{{z}}^{{4}}{+}{235447336673280}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{8}}{}{z}{-}{14674245120000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{7}}{}{{z}}^{{2}}{+}{163796824166400}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{}{{z}}^{{3}}{-}{37324800000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{{z}}^{{6}}{+}{3675750562759680}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{7}}{}{z}{+}{227541778560000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{}{{z}}^{{2}}{+}{66355200000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{}{{z}}^{{4}}{-}{4534963200000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{{z}}^{{5}}{+}{5039700480000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{5}}{}{{z}}^{{2}}{-}{129548782080000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{}{{z}}^{{3}}{+}{21017180160000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{{z}}^{{4}}{-}{629856000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{{z}}^{{5}}{-}{1018259361792000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{5}}{}{z}{+}{347786987328000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{}{{z}}^{{2}}{+}{2203937994240000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{{z}}^{{3}}{+}{139968000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{}{{z}}^{{5}}{-}{18743464800000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{{z}}^{{4}}{+}{18852048082512000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{{z}}^{{2}}{+}{4076179200000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{2}}{}{{z}}^{{3}}{+}{58320000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{}{{z}}^{{4}}{+}{94478400000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{{z}}^{{3}}{-}{32611248000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{2}}{}{{z}}^{{2}}{-}{528160248000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{}{{z}}^{{3}}{+}{27199544370884352}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{+}{2614664021875200}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{7}}{+}{321705392979406400}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{-}{23046304810528800}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{-}{104840997530040000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{2}}{+}{85139930065725000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{z}}^{{2}}{-}{546360629280768}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{11}}{+}{1967799836731392}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{10}}{-}{7519204604620800}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{8}}{-}{209952000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{z}}^{{6}}{-}{39416496242604800}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{5}}{-}{15446376000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{z}}^{{5}}{-}{2927664000000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{z}}^{{4}}{+}{6063266599200000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{z}}^{{3}}{-}{14135648256000}{}{{y}}^{{9}}{}{{z}}^{{3}}{-}{175841686425600}{}{{y}}^{{6}}{}{{z}}^{{3}}{-}{3045221256960000}{}{{y}}^{{3}}{}{{z}}^{{3}}{-}{737167716000000}{}{x}{}{{z}}^{{3}}{-}{359455142707200}{}{{y}}^{{10}}{}{z}{+}{257997550049280}{}{{y}}^{{7}}{}{z}{-}{25470717458112000}{}{{y}}^{{4}}{}{z}{+}{300987187200}{}{{y}}^{{9}}{}{{z}}^{{4}}{-}{725594112000}{}{{y}}^{{6}}{}{{z}}^{{5}}{-}{7759825920000}{}{{y}}^{{6}}{}{{z}}^{{4}}{-}{37324800000}{}{{y}}^{{3}}{}{{z}}^{{6}}{-}{7054387200000}{}{{y}}^{{3}}{}{{z}}^{{5}}{+}{46656000000}{}{x}{}{{z}}^{{6}}{-}{209844224640000}{}{{y}}^{{3}}{}{{z}}^{{4}}{+}{139968000000}{}{y}{}{{z}}^{{5}}{-}{38688904800000}{}{x}{}{{z}}^{{4}}{-}{8777828946600000}{}{{z}}^{{2}}{}{y}{-}{8668550430720}{}{{y}}^{{10}}{}{{z}}^{{2}}{-}{39028901068800}{}{{y}}^{{7}}{}{{z}}^{{3}}{-}{42639851520000}{}{{y}}^{{7}}{}{{z}}^{{2}}{-}{213580800000}{}{{y}}^{{4}}{}{{z}}^{{4}}{-}{380772679680000}{}{{y}}^{{4}}{}{{z}}^{{3}}{-}{2230302304512000}{}{{y}}^{{4}}{}{{z}}^{{2}}{+}{11844403200000}{}{{y}}^{{2}}{}{{z}}^{{3}}{+}{6298560000000}{}{y}{}{{z}}^{{4}}{-}{696965508000000}{}{y}{}{{z}}^{{3}}{-}{1409502931845120}{}{{y}}^{{8}}{}{z}{-}{2537233920000}{}{{y}}^{{5}}{}{{z}}^{{2}}{-}{18392297260032000}{}{{y}}^{{5}}{}{z}{-}{49118173405280000}{}{{y}}^{{2}}{}{z}{-}{870738012000000}{}{{z}}^{{4}}{-}{17937936000000}{}{{z}}^{{5}}{-}{7353752910796800}{}{{y}}^{{8}}{-}{26046898913260800}{}{{y}}^{{5}}{-}{568616554967464800}{}{{y}}^{{4}}{-}{1139103470665728}{}{{y}}^{{11}}{-}{187882578562680000}{}{y}{}{z}{-}{3875300052120000}{}{x}{}{{z}}^{{2}}{+}{39860784187015680}{}{{y}}^{{6}}{}{z}{+}{32041679675265792}{}{{y}}^{{9}}{+}{88088744959114400}{}{{y}}^{{6}}{-}{17319759203700000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{109342506627225000}{}{{z}}^{{2}}{+}{2118938395306196250}{}{z}{+}{991358206562039625}{}{x}{-}{1534072805076652500}{}{y}{+}{597219742017708750}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{z}{+}{94086819258781500}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{-}{200252907899415000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{+}{94950792000000}{}{{y}}^{{2}}{}{{z}}^{{2}}{-}{524766413168640}{}{{y}}^{{9}}{}{{z}}^{{2}}{-}{1314534666240000}{}{{y}}^{{6}}{}{{z}}^{{2}}{-}{12373762321800000}{}{{z}}^{{3}}{+}{682176021898628000}{}{{y}}^{{3}}{}{z}{+}{163552851034668000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{z}{+}{38444953961075000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{z}{-}{7560832848058368}{}{{y}}^{{10}}{-}{107487335683180800}{}{{y}}^{{7}}{-}{307559853657459000}{}{{y}}^{{3}}{-}{22027797731340000}{}{{y}}^{{2}}{+}{84698873079075000}{}{x}{}{z}{-}{670150659686400}{}{{y}}^{{9}}{}{z}{+}{31476478240152000}{}{{y}}^{{3}}{}{{z}}^{{2}}{+}{841574249783608500}{}{{y}}^{{3}}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){+}{48514014178143750}{,}{159225750}{-}{3596400}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{{z}}^{{2}}{+}{121150080}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{}{z}{-}{39096000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{}{z}{+}{10425600}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{{z}}^{{2}}{+}{40326912}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{9}}{-}{51710400}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{-}{460252800}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{4}}{+}{42460200}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{z}}^{{2}}{-}{378064800}{}{{y}}^{{4}}{+}{14850000}{}{y}{}{z}{-}{13032000}{}{x}{}{{z}}^{{2}}{+}{78382080}{}{{y}}^{{6}}{}{z}{-}{205141248}{}{{y}}^{{9}}{-}{2779336800}{}{{y}}^{{6}}{+}{92534400}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){-}{80919000}{}{{z}}^{{2}}{-}{2853557750}{}{z}{+}{238838625}{}{x}{+}{22477500}{}{y}{+}{3396141950}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{z}{+}{1702428300}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{-}{2617839000}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{-}{246564000}{}{{y}}^{{3}}{}{z}{+}{1331445600}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{z}{+}{42460200}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{z}{-}{7894611000}{}{{y}}^{{3}}{-}{80919000}{}{x}{}{z}{-}{3960000}{}{{y}}^{{3}}{}{{z}}^{{2}}{+}{579121000}{}{y}{}{x}{-}{1130633900}{}{{y}}^{{3}}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){,}{-}{896}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{6}}{-}{736}{}{{y}}^{{6}}{-}{80}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{{y}}^{{3}}{}{z}{-}{4860}{}{{y}}^{{3}}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){-}{2240}{}{{y}}^{{3}}{}{z}{-}{540}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{}{z}{+}{900}{}{{y}}^{{3}}{-}{1440}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{x}{+}{300}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{y}{-}{2880}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){}{z}{+}{7610}{}{{x}}^{{2}}{+}{100}{}{x}{}{z}{-}{960}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{5}\right){-}{6075}{}{x}{+}{8400}{}{y}{-}{12150}{}{z}{-}{4050}\right]$ (6)

References

 Amrhein, B.; Gloor, O.; and Kuchlin, W. "On the Walk." Theoretical Comput. Sci., Vol. 187, (1997): 179-202.
 Collart, S.; Kalkbrener, M.; and Mall, D. "Converting Bases with the Grobner Walk." J. Symbolic Comput., Vol. 3, No. 4, (1997): 465-469.
 Tran, Q.N. "A Fast Algorithm for Grobner Basis Conversion and Its Applications." J. Symbolic Comput., Vol. 30, (2000): 451-467.