This functions uses the F4 implementation in the msolve package to compute a Groebner basis, in GRevLex order, of a polynomial ideal with either rational coefficients or finite field coefficients with characteristic less than $2^{31}$. If the input ideal is a polynomial ring with monomial order other than GRevLex a GRevLex basis is returned (and no warning is given). The input ideal may also be given in a ring with integer coefficients, in this case a Groebner basis for the given ideal over the rationals will be computed, denominators will be cleared, and the output will be a Groebner basis over the rationals in GRevLex order with integer coefficients.
i1 : R=ZZ/1073741827[z_1..z_3]
o1 = R
o1 : PolynomialRing
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i2 : I=ideal(7*z_1*z_2+5*z_2*z_3+z_3^2+z_1+5*z_3+10,8*z_1^2+13*z_1*z_3+10*z_3^2+z_2+z_1)
2 2 2
o2 = ideal (7z z + 5z z + z + z + 5z + 10, 8z + 13z z + 10z + z +
1 2 2 3 3 1 3 1 1 3 3 1
------------------------------------------------------------------------
z )
2
o2 : Ideal of R
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i3 : gB=msolveGB I
o3 = | z_1z_2-460175068z_2z_3-306783379z_3^2-306783379z_1-460175068z_3+
------------------------------------------------------------------------
153391691 z_1^2+134217730z_1z_3+268435458z_3^2-402653185z_1-402653185z_2
------------------------------------------------------------------------
z_1z_3^2+19173957z_2z_3^2-479349029z_3^3-402653186z_2^2+460175073z_1z_3+
------------------------------------------------------------------------
134217729z_2z_3+153391693z_3^2+10z_1+402653185z_2-364305253z_3+268435458
------------------------------------------------------------------------
z_2^2z_3^2+59398484z_2z_3^3+347252676z_3^4+516309900z_2^3+91382283z_2^2z
------------------------------------------------------------------------
_3+27414685z_2z_3^2+223886594z_3^3-466049644z_2^2-68536713z_2z_3+
------------------------------------------------------------------------
310699764z_3^2-310699763z_2+411220277z_3-68536709 |
1 4
o3 : Matrix R <-- R
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i4 : lT=monomialIdeal leadTerm gB
2 2 2 2
o4 = monomialIdeal (z , z z , z z , z z )
1 1 2 1 3 2 3
o4 : MonomialIdeal of R
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i5 : degree lT
o5 = 4
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i6 : dim lT
o6 = 1
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i7 : R=QQ[z_1..z_3]
o7 = R
o7 : PolynomialRing
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i8 : I=ideal(7*z_1*z_2+5*z_2*z_3+z_3^2+z_1+5*z_3+10,8*z_1^2+13*z_1*z_3+10*z_3^2+z_2+z_1)
2 2 2
o8 = ideal (7z z + 5z z + z + z + 5z + 10, 8z + 13z z + 10z + z +
1 2 2 3 3 1 3 1 1 3 3 1
------------------------------------------------------------------------
z )
2
o8 : Ideal of R
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i9 : gB=msolveGB I
o9 = | 7z_1z_2+5z_2z_3+z_3^2+z_1+5z_3+10 8z_1^2+13z_1z_3+10z_3^2+z_1+z_2
------------------------------------------------------------------------
56z_1z_3^2-235z_2z_3^2+51z_3^3-49z_2^2+240z_1z_3+35z_2z_3+192z_3^2+560z_
------------------------------------------------------------------------
1-7z_2+545z_3+70 235z_2^2z_3^2-11z_2z_3^3+8z_3^4+49z_2^3-35z_2^2z_3+13z_
------------------------------------------------------------------------
2z_3^2+67z_3^3+14z_2^2-150z_2z_3+304z_3^2-69z_2+665z_3+790 |
1 4
o9 : Matrix R <-- R
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i10 : (ideal gB)== ideal(groebnerBasis I)
o10 = true
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i11 : lT=monomialIdeal leadTerm gB
2 2 2 2
o11 = monomialIdeal (z , z z , z z , z z )
1 1 2 1 3 2 3
o11 : MonomialIdeal of R
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i12 : degree lT
o12 = 4
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i13 : dim lT
o13 = 1
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