Reduced Gröbner Basis for $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$

Question

I have $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$ with lex $X>Y>Z $.

I have calculated the Gröbner Basis as $G=\{ X^2+2XYZ, XY+2Y^2Z-1, X, 2Y^2Z-1 \}$. But the question I have asks for the Reduced Gröbner Basis and I'm not really sure how I go about finding it. I think the minimal Gröbner Basis is: $\{X, 2Y^2Z-1 \}$.

Answer 1

According to Wikipedia, "A Gröbner basis is termed reduced if the leading coefficient of each element of the basis is 1 and no monomial in any element of the basis is in the ideal generated by the leading terms of the other elements of the basis."

So, the first property isn't true for your last basis element. 2XYZ is generated by XY, so you couldn't have both $X^2+2XYZ$ and $XY+2Y^2Z$.