RememberBasis - Maple Help

Groebner

 RememberBasis
 make a Groebner basis known to the system

 Calling Sequence RememberBasis(F, G, T, characteristic=p)

Parameters

 F - a list or set of polynomials or a PolynomialIdeal G - a Groebner basis or a name T - ShortMonomialOrder or MonomialOrder p - (optional) characteristic: either zero or a prime

Description

 • RememberBasis marks the list of polynomials G as a Groebner basis of the ideal generated by F with respect to the monomial order T.  This allows you to use Groebner bases computed outside of Maple or saved in a previous Maple session. The optional argument characteristic=p can be used to specify the ring characteristic when T is a ShortMonomialOrder.  The default characteristic is zero.
 • If G is not a reduced Groebner basis you can use the InterReduce command to reduce it first.  This is also advisable if the polynomials are not in Maple's canonical form, (for example, if they are monic).  Maple stores Groebner bases as polynomials that are primitive and fraction-free.
 • If a different Groebner basis is already known for a particular problem then RememberBasis will replace it with G. If G is a name then RememberBasis will forget any basis that is known.
 • Note that the pretend_gbasis command is deprecated.  It may not be supported in a future Maple release.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[{x}^{2}yz+x{y}^{2}z+xy{z}^{2}+xyz+xy+xz+yz,{x}^{2}{y}^{2}z+{x}^{2}yz+x{y}^{2}{z}^{2}+xyz+x+yz+z,{x}^{2}{y}^{2}{z}^{2}+{x}^{2}{y}^{2}z+x{y}^{2}z+xyz+xz+z+1\right]$
 ${F}{≔}\left[{{x}}^{{2}}{}{y}{}{z}{+}{x}{}{{y}}^{{2}}{}{z}{+}{x}{}{y}{}{{z}}^{{2}}{+}{x}{}{y}{}{z}{+}{x}{}{y}{+}{x}{}{z}{+}{y}{}{z}{,}{{x}}^{{2}}{}{{y}}^{{2}}{}{z}{+}{x}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{{x}}^{{2}}{}{y}{}{z}{+}{x}{}{y}{}{z}{+}{y}{}{z}{+}{x}{+}{z}{,}{{x}}^{{2}}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{{x}}^{{2}}{}{{y}}^{{2}}{}{z}{+}{x}{}{{y}}^{{2}}{}{z}{+}{x}{}{y}{}{z}{+}{x}{}{z}{+}{z}{+}{1}\right]$ (1)
 > $G≔\left[-42{x}^{4}-431{x}^{3}-757{x}^{2}y+804x{y}^{2}-59{y}^{3}-799{x}^{2}z+2xyz+119{y}^{2}z-417x{z}^{2}+340y{z}^{2}-5{z}^{3}-303{x}^{2}+1203xy-194{y}^{2}+752xz-246yz-184{z}^{2}-581x+67y-1013z+57,21{x}^{3}y-16{x}^{3}-74{x}^{2}y+24x{y}^{2}+2{y}^{3}-53{x}^{2}z+xyz-35{y}^{2}z+12x{z}^{2}+23y{z}^{2}+8{z}^{3}-36{x}^{2}+24xy+29{y}^{2}+40xz+3yz-8{z}^{2}-28x+23y-13z+18,-21{x}^{2}{y}^{2}-2{x}^{3}-25{x}^{2}y+45x{y}^{2}-5{y}^{3}-25{x}^{2}z+29xyz+14{y}^{2}z-9x{z}^{2}+16y{z}^{2}+{z}^{3}+6{x}^{2}+45xy-20{y}^{2}+47xz-18yz-{z}^{2}-14x-5y-41z-3,42x{y}^{3}-11{x}^{3}-85{x}^{2}y+6x{y}^{2}+25{y}^{3}-43{x}^{2}z-40xyz-49{y}^{2}z-39x{z}^{2}+4y{z}^{2}-5{z}^{3}-51{x}^{2}-15xy+58{y}^{2}-4xz+6yz-16{z}^{2}-35x+25y-5z+15,42{y}^{4}-121{x}^{3}-557{x}^{2}y+570x{y}^{2}+275{y}^{3}-515{x}^{2}z-608xyz-77{y}^{2}z-555x{z}^{2}+2y{z}^{2}-55{z}^{3}-645{x}^{2}+633xy-160{y}^{2}+82xz-690yz-302{z}^{2}-679x+65y-1147z-3,-7{x}^{3}z+5{x}^{3}+24{x}^{2}y-25x{y}^{2}+2{y}^{3}+17{x}^{2}z+8xyz-7{y}^{2}z+19x{z}^{2}-12y{z}^{2}+{z}^{3}+6{x}^{2}-32xy+8{y}^{2}-23xz+17yz+6{z}^{2}+21x+2y+36z-3,14{x}^{2}yz+3{x}^{3}+13{x}^{2}y+6x{y}^{2}-3{y}^{3}-{x}^{2}z+2xyz+21{y}^{2}z-25x{z}^{2}+4y{z}^{2}-5{z}^{3}+19{x}^{2}+27xy-26{y}^{2}+10xz-8yz-2{z}^{2}-7x-17y-33z-13,-6x{y}^{2}z+{x}^{3}+5{x}^{2}y-12x{y}^{2}+{y}^{3}+11{x}^{2}z+2xyz-7{y}^{2}z+9x{z}^{2}-2y{z}^{2}+{z}^{3}+3{x}^{2}-15xy+4{y}^{2}-4xz+6yz+2{z}^{2}+13x+y+19z+3,-21{y}^{3}z+4{x}^{3}-13{x}^{2}y-48x{y}^{2}+10{y}^{3}+8{x}^{2}z+5xyz+14{y}^{2}z-3x{z}^{2}+10y{z}^{2}-2{z}^{3}+30{x}^{2}-27xy+40{y}^{2}-10xz+57yz+2{z}^{2}+7x+10y+40z+6,14{x}^{2}{z}^{2}+5{x}^{3}+31{x}^{2}y-74x{y}^{2}+9{y}^{3}+45{x}^{2}z+22xyz-35{y}^{2}z+61x{z}^{2}-40y{z}^{2}+{z}^{3}+13{x}^{2}-95xy+50{y}^{2}-44xz+66yz+6{z}^{2}+63x+23y+127z+11,21xy{z}^{2}-{x}^{3}-2{x}^{2}y-51x{y}^{2}+8{y}^{3}+40{x}^{2}z+25xyz-56{y}^{2}z+69x{z}^{2}-13y{z}^{2}+11{z}^{3}-18{x}^{2}-72xy+53{y}^{2}-8xz+54yz+10{z}^{2}+56x+29y+116z+30,14{y}^{2}{z}^{2}+{x}^{3}+9{x}^{2}y+2x{y}^{2}-{y}^{3}+23{x}^{2}z+10xyz-21{y}^{2}z+15x{z}^{2}-8y{z}^{2}+3{z}^{3}-3{x}^{2}-19xy-4{y}^{2}-6xz-26yz-10{z}^{2}+7x-y+3z+5,21x{z}^{3}-29{x}^{3}-121{x}^{2}y+369x{y}^{2}-41{y}^{3}-226{x}^{2}z-178xyz+161{y}^{2}z-267x{z}^{2}+148y{z}^{2}+4{z}^{3}-123{x}^{2}+411xy-206{y}^{2}+146xz-324yz+38{z}^{2}-329x-62y-584z-75,21y{z}^{3}-10{x}^{3}-41{x}^{2}y+183x{y}^{2}-25{y}^{3}-104{x}^{2}z-65xyz+91{y}^{2}z-129x{z}^{2}+59y{z}^{2}-16{z}^{3}-33{x}^{2}+225xy-142{y}^{2}+67xz-174yz-5{z}^{2}-154x-46y-310z-57,323{y}^{3}+4799z+281y+2849x-1100xz+2574yz-3495xy+1642xyz+1153{x}^{2}y-3054x{y}^{2}+1544{y}^{2}+1347{x}^{2}-368{z}^{2}-1211{y}^{2}z-1252y{z}^{2}+2253x{z}^{2}-31{z}^{3}+2035{x}^{2}z+42{z}^{4}+293{x}^{3}+555\right]:$

Assume that G is known to be a Groebner basis for F with respect to tdeg(x,y,z). We will make this basis known to the system, so that it is not recomputed.

 > $\mathrm{RememberBasis}\left(F,G,\mathrm{tdeg}\left(x,y,z\right)\right)$
 > $\mathrm{time}\left(\mathrm{HilbertDimension}\left(F,\mathrm{tdeg}\left(x,y,z\right)\right)\right)$
 ${0.001}$ (2)