Chapter 15

Write the indicated polynomials in \(\mathbb{R}[x, y, z]\) in decreasing term order using the lexicographic order with \(x>y>z\). $$ 3 x y-5 y z+7 x z $$

Write the indicated polynomials in \(\mathbb{R}[x, y, z]\) in decreasing term order using the lexicographic order with \(x>y>z\). $$ 3 z-2 x+y^{2}-z^{2}+x y $$

Write the indicated polynomials in \(\mathbb{R}[x, y, z]\) in decreasing term order using the lexicographic order with \(x>y>z\). $$ 5+3 x^{2} z-2 x y^{4} z^{3}+3 z-5 x+2 y $$

Write the indicated polynomials in \(\mathbb{R}[x, y, z]\) in decreasing term order using the lexicographic order with \(x>y>z\). $$ x y z^{4}-x y^{2} z+x^{2} y z+x^{3} z-x^{5} $$

Let \(f, g \in F\left[x_{1}, \ldots, x_{n}\right]\) be nonzero polynomials. Show that if \(f+g \neq 0,\) then multideg \((f+g) \leq \max \\{\) multideg \(f,\) multideg \(g\\}\)

Calculate the remainder on dividing \(f\) by the given sets of polynomials \(S\), using the lexicographic order. $$ f=x^{2} y z+x z^{2}-y z \quad S=\left\\{x^{2}-y, y-z\right\\} $$

Calculate the remainder on dividing \(f\) by the given sets of polynomials \(S\), using the lexicographic order. $$ f=x^{3} y^{2}-x y z+y z^{2} \quad S=\left\\{x^{2}-y z, x+z^{2}, y-z\right\\} $$

Calculate the \(S\) -polynomials \(S(f, g)\) of the indicated polynomials \(f, g\) using the lexicographic order with \(x>y>z\) $$ f=x y-z, g=x^{2}+y z $$

Calculate the \(S\) -polynomials \(S(f, g)\) of the indicated polynomials \(f, g\) using the lexicographic order with \(x>y>z\) $$ f=x y^{2}+z^{4}, g=x^{2} y-z^{2} $$

Calculate the \(S\) -polynomials \(S(f, g)\) of the indicated polynomials \(f, g\) using the lexicographic order with \(x>y>z\) $$ f=x^{4} z-y^{2}, g=x y^{2}-z $$

Calculate the \(S\) -polynomials \(S(f, g)\) of the indicated polynomials \(f, g\) using the lexicographic order with \(x>y>z\) $$ f=x y^{2} z+3 x y^{4}, g=x^{2} y-z^{2} $$

Calculate the \(S\) -polynomials \(S(f, g)\) of the indicated polynomials \(f, g\) using the lexicographic order with \(x>y>z\) $$ f=x^{3} y^{2} z-x+y, g=x^{2} z^{3}+z $$

Construct a Gröbner basis for the following ideals in \(\mathbb{R}[x, y, z]\) with \(x>y>z\) $$ I=\langle x-y, x+y\rangle $$

Construct a Gröbner basis for the following ideals in \(\mathbb{R}[x, y, z]\) with \(x>y>z\) $$ I=\langle x y-z, x-y z\rangle $$

Construct a Gröbner basis for the following ideals in \(\mathbb{R}[x, y, z]\) with \(x>y>z\) $$ I=\langle x-y+z, x+y-2 z, 3 x-y+3 z\rangle $$

Construct a Gröbner basis for the following ideals in \(\mathbb{R}[x, y, z]\) with \(x>y>z\) $$ I=\left\langle x^{2} y-x y^{2}, x y-x\right\rangle $$