Math and stuff

Proposition If we use grlex order with $x > y > z$, is $\{ x^4y^2 - z^5, x^3y^3 - 1, x^2y^4 - 2z \}$ a Grobner basis for the ideal generated by these polynomials? Solution It is not. Let $g_1, g_2, g_3$ denote the three polynomials. $2xz - y =...

Proposition Let $f \in k[x_1, \cdots, x_n]$. If $f \notin \ev{ x_1, \cdots, x_n }$, then show $\ev{ x_1, \cdots, x_n, f } = k[x_1, \cdots, x_n]$. Solution Since $f \in k[x_1, \cdots, x_n]$, $f = c_{\alpha_1} x^{\alpha_1} + \cdots + c_{\alpha_n} x^{\alpha_n}$ for some $\alpha_i$ and nonzero $c_{\alpha_i}$. For...

Proposition If $I = \ev{ x^{\alpha(1)}, \cdots, x^{\alpha(s)} }$ is a monomial ideal, prove that a polynomial $f$ is in $I$ if and only if the remainder of $f$ on division by $x^{\alpha(1)}, \cdots, x^{\alpha(s)}$ is zero. Solution If the remainder of $f$ on division by $x^{\alpha(1)}, \cdots, x^{\alpha(s)}$ is...

Proposition Let $I = \ev{ x^6, x^2y^3, xy^7 } \subset k[x, y]$. In the $(m, n)$-lane, plot the set of exponent vectors $(m, n)$ of monomials $x^my^n$ appearing in elements of $I$. If we apply the division algorithm to an element $f \in k[x, y]$, using the generators of $I$...

Proposition Let $I$ be a monomial ideal and $f \in k[x_1, \cdots, x_n]$. If $f \in I$, then every term of $f$ lies in $I$. Solution By the definition of a monomial ideal, there is a subset $A \subset \mathbb{Z}^n_{\geq 0}$ such that $I$ consists of all polynomials which are...