Math and stuff

Proposition Prove that $\{ 0 \}$ and $k$ are the only ideals of a field $k$. Solution Let $I \ne \{ 0 \}$ be an ideal of $k$. Such an ideal must exist since $k$ is an ideal of $k$. Let $x \in I$ such that $x \ne 0$. Since...

Proposition Let $J = \ev{ x^2 + y^2 - 1, y - 1 }$. Find $f \in I(V(J))$ such that $f \notin J$. Solution GroebnerBasis([x**2, y - 1], x, y, domain='ZZ', order='lex') Sympy shows that $\{ x^2, y - 1 \}$ is a Groebner basis of $J$. Thus $(x, y)...

Proposition Determine whether $f = x^3z - 2y^2$ is in the ideal $I = \ev{ xz - y, xy + 2z^2, y - z }$. Solution from sympy import * from sympy.polys.orderings import monomial_key x, y, z = symbols('x y z') print(groebner([x * z - y, x * y +...

Proposition Find the points in $\mathbb{C}^3$ on the variety \[\begin{align*} V(x^2 + y^2 + z^2 - 1, x^2 + y^2 + z^2 - 2x, 2x - 3y - z). \end{align*}\] Solution from sympy import * from sympy.polys.orderings import monomial_key x, y, z = symbols('x y z') print(groebner([x**2 + y**2 +...

Proposition If $G = \{ g_1, \cdots, g_k \}$ is a basis for $I$ with the property that $\overline{f}^{G} = 0$ for all $f \in I$, then $G$ is a Groebner basis. Solution For each $(i, j)$ pair, $S(g_i, g_j) = \frac{x^{\gamma}}{\LT(g_i)} g_i - \frac{x^{\gamma}}{\LT(g_j)} g_j$ where $x^{\gamma} = \lcm(\LT(g_i),...