In[]:=
Clear[f,g];f=x^3z-2y^2;g={xz-y,xy+2z^2,y-z};Resolve[ForAll[{x,y,z},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