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KK.<x,y>=QQ[]
I=Ideal([x^4-3*x*y+y^2,x^5-4*x*y+3*x*y^2])
︠0c007b62-f9cc-4c6a-9450-7c5ea12e8844︠
x^2 in I
False 
x^3 in I
False 
x^4 - 3*x*y + y^2 in I
True 
I.groebner_basis()
[y^5 - 8*y^4 - 531/16*x*y^2 + 273/16*y^3 + 12*x*y, x^4 - 3*x*y + y^2, x*y^3 + 2/3*y^4 - 4/3*y^3, x^2*y + 2/3*x*y^2 - 4/3*x*y]