Next two cells are from original question. The correct answer is False
In[]:=
Clear[f,g];f=x^3z-2y^2;g={xz-y,xy+2z^2,y-z};​​Resolve[​​ForAll[{x,y,z},​​Evaluate[PolynomialReduce[f,GroebnerBasis[g,{x,y,z}],{x,y,z}]][[-1]]]==0]
Out[]=
True
In[]:=
Clear[f,g,result];f=x^3z-2y^2;g={xz-y,xy+2z^2,y-z};​​result=PolynomialReduce[f,GroebnerBasis[g,{x,y,z}],{x,y,z}][[-1]];​​Resolve[ForAll[{x,y,z},result==0]]
Out[]=
False
Next three cells show Evaluate[] used correctly in the first cell and incorrectly in the next two
In[]:=
Clear[f,g];f=x^3z-2y^2;g={xz-y,xy+2z^2,y-z};​​Resolve[ForAll[{x,y,z},
Evaluate
[
PolynomialReduce[f,GroebnerBasis[g,{x,y,z}],{x,y,z}][[-1]]==0]
]
]
Out[]=
False
In[]:=
Clear[f,g];f=x^3z-2y^2;g={xz-y,xy+2z^2,y-z};​​Resolve[ForAll[{x,y,z},
Evaluate
[
PolynomialReduce[f,GroebnerBasis[g,{x,y,z}],{x,y,z}]
]
[[-1]]==0]]
Out[]=
True
In[]:=
Clear[f,g];f=x^3z-2y^2;g={xz-y,xy+2z^2,y-z};​​Resolve[ForAll[{x,y,z},
Evaluate
[
PolynomialReduce[f,GroebnerBasis[g,{x,y,z}],{x,y,z}][[-1]]
]
==0]]
Out[]=
True