Math and stuff

Proposition Rewrite each of the following polynomials, ordering the terms using the lex order, grlex order, and the grevlex order, giving $\LM(f), \LT(f)$ and $\multideg(f)$ in each case. $f(x, y, z) = 2x + 3y + z + x^2 - z^2 + x^3$. $f(x, y, z) = 2x^2y^8 - 3x^5yz^4...

Proposition Consider the equations \[\begin{align*} x^2 + y^2 - 1 &= 0, \\ xy - 1 &= 0 \end{align*}\] which describe the intersection of a circle and a hyperbola. Use algebra to eliminate $y$ from the above equations. Show how the polynomial found in part (1) lies in $\ev{ x^2...

Proposition Parametrize all solutions to the linear equations \[\begin{align*} x + 2y - 2z + w &= -1, \\ x + y + z - w &= 2. \end{align*}\] Solution From \[\begin{align*} \begin{bmatrix} 1 & 2 & -2 & 1 \\ 1 & 1 & 1 & -1 \end{bmatrix} \begin{bmatrix}...

Proposition Use a trigonometric identity to show that \[\begin{align*} x &= \cos(t), \\ y &= \cos(2t) \end{align*}\] parametrizes a portion of a parabola. Indicate exactly what portion of the parabola is covered. Solution Since $\cos(2t) = 2\cos^2(t) - 1$, $y = 2x^2 - 1$. Since $x, y \in [-1, 1]$,...

Proposition Prove that any two distinct points in $\mathbb{P}^2(\mathbb{R})$ determine a unique projective line. Prove that any two distinct projective lines in $\mathbb{P}^2(\mathbb{R})$ meet at a unique point. Solution We will define $\mathbb{P}^2(\mathbb{R})$ as the set containing $\mathbb{R}^2$ and one point at $\infty$ for each equivalence class of parallel lines....