1. Compute the leading term and leading coefficient of:

$$f = 4xy^2z + 4z^2 -5x^3 + 7xy^2 -7y^4$$

with respect to the orderings:

• 'lex' on $\mathbb Q[x,y,z]$
•  'deglex' on $\left(\mathbb Z / 2\mathbb Z\right)[z,y,x]$

2. Determine matrices defining the orderings (for $3$ variables): 'lex', 'neglex', 'degrevlex', 'deglex', 'negdegrevlex' and 'negdeglex'.

3. Give one possible realization of the following rings in SAGE:

• \item $\mathbb Q[x,y,z]$
  \item $\mathbb F_5[y,x,z]$
  \item $\mathbb Q[x,y,z] / \langle x^5 + y^3 + z^2\rangle$
  \end{column}
  \begin{column}{4cm}
  \item $\left(\mathbb Z / \mathbb Z_2\right)[x_1, \ldots, x_{10}]$
  \item $\mathbb F_5[x,y,z]_{\langle x,y,z\rangle}$
  \item $\mathbb Q[x,y,z]_{\langle x,y \rangle}$
  $\mathbb Q[x,y,z]$
• $\mathbb F_5[y,x,z]$
• $\mathbb Q[x,y,z] / \langle x^5 + y^3 + z^2\rangle$
• $\left(\mathbb Z / \mathbb Z_2\right)[x_1, \ldots, x_{10}]$
• $\mathbb F_5[x,y,z]_{\langle x,y,z\rangle}$
• $\mathbb Q[x,y,z]_{\langle x,y \rangle}$

HINT: Let $\succ$ be a local ordering on $\mathbb K[x_1, \ldots, x_n]$ then: $$\mathbb K[x_1, \ldots, x_n, y_1, \ldots, y_m] = \mathbb K(y_1, \ldots, y_m) [x_1, \ldots, x_n]_{\succ}.$$

4. Which of the following orderings are elimination orderings: 'lex', 'neglex' or '(lex(n), neglex(m))'.

Compute a standard basis of the ideal $$\langle x-t^2, y-t^3, z-t^4\rangle$$ for all those orderings.

5. Obtain an standard basis of the ideal $\mathcal I\cap \mathbb K[x,y,z]$ where $\mathcal I = \langle t^2 + x^2 + y^2+z^2, t^2+2x^2-xy-z^2\rangle\subset \mathbb K[x,y,z,t]$.

6. Check whether the following polynomials are contained in the ideal $\mathcal I =\langle x^{10}+x^9y^2, y^8-x^2y^7\rangle$ of the ring $\mathbb Q[x,y,z]$ and the local ring $\mathbb Q[x,y,z]_{\langle x,y,z\rangle}$:

$$\begin{array}{c}f_1=x^2y^7+y^{14} \\ f_2=xy^{13}+y^{12}\end{array}$$

7. Use SAGE to solve the following linear system of equation:

$$\begin{array}{c}

x+5y=2 \\

-3x+6y=15

\end{array}$$

Compared the standard basis algorithm with the Gaussian elimination algorithm in this case. Try also the procedure {\tt solve}.

8. Apply the corresponding procedures from SAGE to check whether an ideal or a polynomial is homogeneous and to compute its homogenization.