#### UserFunction GroebnerBasis

GroebnerBasis(u:DPOLID(C,v,O))

The single argument $u$ has the type $DPOLID$ which abbreviates
Distributive Polynomial Ideal. The meaning of its parameters is as follows.

$C$: Coefficient type.

$v=\{v_1,v_2,\ldots \}$: Distributive variables in decreasing order.

$O:LEX|GRLEX|GREVLEX$: Term ordering.

Specification. The reduced Groebner basis of $u$ in the specified term ordering is returned.

Examples. The input is given in Reduce algebraic mode syntax. The output for each
minisession is returned in a separate window.

ps:={y**4-x**3,x*y**3-x**3*y,x**2*y**2-x**2*y,

x**4*y-x**4,x**5-x**4};

GroebnerBasis(ps|DPOLID(Q,{y,x},LEX)|);

Generating Set for Ideal:
$\{y^{4}-x^{3},y^{3}x-yx^{3},y^{2}x^{2}-yx^{2},yx^{4}-x^{4},x^{5}-x^{4}\}$
Groebner Basis for Term Order $LEX$, $y>x$.
$\{y^{4}-x^{3},y^{3}x-x^{4},yx^{2}-x^{4},x^{5}-x^{4}\}$

GroebnerBasis(ps|DPOLID(Q,{x,y},LEX)|);

Generating Set for Ideal:
$\{-x^{3}+y^{4},-x^{3}y+xy^{3},x^{2}y^{2}-x^{2}y,x^{4}y-x^{4},x^{5}-x^{4}\}$
Groebner Basis for Term Order $LEX, x>y$.
$\{x^{3}-y^{4},x^{2}y-y^{5},xy^{3}-y^{5},x^{5}-x^{4}\}$

GroebnerBasis}(u:MODULE(DPOLY(C,vs,O),ORD))

The meaning of the parameters for $DPOLY$ is the same as for $DPOLID$ above.

$ORD=TOP|POT$: Module term ordering.

Examples.

fs:={{0,y,x},{0,x,x*y-x},{x,y**2,0},{y,0,x}};

GroebnerBasis(fs|MODULE(DPOLY(Q,{y,x},GRLEX),TOP)|);

Generating Set for Module:
$\left(\begin{array}{c}0\\y\\x\end{array}\right) \left(\begin{array}{c}0\\x\\xy-x\end{array}\right) \left(\begin{array}{c}x\\y^{2}\\0\end{array}\right) \left(\begin{array}{c}y\\0\\x\end{array}\right)$
Groebner Basis for Term Order $GRLEX,x>y,TOP$
$\left(\begin{array}{c}x\\0\\-xy\end{array}\right)$$\left(\begin{array}{c}y\\0\\x\end{array}\right) \left(\begin{array}{c}0\\x\\xy-x\end{array}\right)$$\left(\begin{array}{c}0\\y\\x \end{array}\right)$$\left(\begin{array}{c}0\\0\\x^{2}+ \frac{1}{2}xy\end{array}\right) \left(\begin{array}{c}0\\0\\xy^{2}- \frac{1}{2}xy\end{array}\right) GroebnerBasis(fs|MODULE(DPOLY(Q,{x,y},LEX),TOP)|); Generating Set for Module: \left(\begin{array}{c}0\\y\\x\end{array}\right) \left(\begin{array}{c}0\\x\\yx-x\end{array}\right) \left(\begin{array}{c}x\\y^{2}\\0\end{array}\right) \left(\begin{array}{c}y\\0\\x\end{array}\right) Groebner Basis for Term Order LEX,x>y,TOP \left(\begin{array}{c}y\\0\\x\end{array}\right) \left(\begin{array}{c}x\\0\\2x^{2} \end{array}\right)$$\left(\begin{array}{c}0\\y\\x\end{array}\right)$ $\left(\begin{array}{c}0\\x\\-2x ^{2}-x\end{array}\right)$$\left(\begin{array}{c}0\\0\\yx+2x^{2} \end{array}\right)$ $\left(\begin{array}{c}0\\0\\x^{3}+ \frac{1}{4}x^{2}\end{array}\right)$

References.

W.W. Adams, P. Loustaunau, An Introduction to Groebner Bases, American Mathematical Society, 1994.

D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms Using Algebraic Geometry,
Undergraduate Texts in Mathematics, Springer, 1992 and 1998.

G.M. Greuel, G. Pfister, A Singular Introduction to Commutative Algebra, Springer, 2002.

M. Kreuzer, L. Robbiano, Using Computational Commutative Algebra I and II, Springer, 2000.