Math and stuff

Proposition Determine whether $f = xy^3 - z^2 + y^5 - z^3$ is in the ideal $I = \ev{ -x^3 + y, x^2y - z }$. Solution from sympy import * from sympy.polys.orderings import monomial_key x, y, z = symbols('x y z') print(groebner([-x**3 + y, x**2 * y - z],...

Proposition Calculate $S(f_i, f_j)$ for each $i < j$. (Grlex) $f_1 = x^3 - 2xy$. $f_2 = x^2y - 2y^2 + x$. $f_3 = -x^2$. $f_4 = -2xy$. $f_5 = -2y^2 + x$. Solution $S(f_1, f_2) = -x^2$. $S(f_1, f_3) = -2xy$. $S(f_1, f_4) = -2xy^2 = yf_4$. $S(f_1, f_5)...

Proposition Fix a monomial ordering and let $I \subset k[x_1, \cdots, x_n]$. Suppose that $f \in k[x_1, \cdots, x_n]$. Show that $f$ can be written in the form $f = g + r$ where $g \in I$ and no term of $r$ is divisible by any element of $\LT(I)$. Given...

Section 2.5 of Ideals, Varieties, and Algorithms shows that $G = \{ x + z, y - z \}$ is a Groebner basis for lex order. Dividing $xy$ by $x + z, y - z$ shows that $xy = y(x + z) - z(y - z) - z^2$. Dividing $xy$...

Problem statement Each of the following polynomials is written with its monomials ordered according to (exactly) one of lex, grlex, or grevlex order. Determine which monomial order was used in each case. $f(x, y, z) = 7x^2y^4z - 2xy^6 + x^2y^2$. $f(x, y, z) = xy^3z + xy^2z^2 + x^2z^3$....