#### UserFunction Quotient

Quotient}(u,v:DPOLID(C,vs,O))

The arguments $u$ and $v$ have the type $DPOLID$ which abbreviates Distributive Polynomial Ideal

The meaning of its parameters is as follows.

$C$: Type of coefficients.

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

$O=LEX|GRLEX|GREVLEX$: Term order.

Specification. The ideal quotient of the ideals $u$ and $v$ is returned.

Examples. The input for the examples is given in Reduce algebraic mode syntax.

The output for each minisession is returned in a separate window.

The first example is from Adams and Loustaunau, page 73.

I:={x**2,x+y}; J:={x*(x+y)**2,y};

T==|DPOLID(Q,{x,y},GRLEX)|;

Quotient(I,J|T|);

The given ideals:
$<x^2,x+y>, <x(x+y)^2,y>$
The quotient ideal:
$<x,y>$

The next example is more complicated.

f:={x*y+2*x-3,x**2+3*x+y}; g:={3*x*y-x+y**2-1,x+3*y-1};

s:=Quotient(f,g|DPOLID(Q,{x,y},GRLEX)|);

The given ideals:
$<xy+2x-3,x^{2}+3x+y>$
$>3xy+y^{2}-x-1,x+3y-1>$
The quotient ideal:
$< y^{5}+\frac{13}{4}y^4+\frac{41}{4}y^3+\frac{73}{4}y^2-17y+\frac{27}{4},$

$x+\frac{22592}{201333}y^{4}+\frac{63388}{201333}y^{3} + \frac{252887}{201333}y^{2}+ \frac{393466}{201333}y -\frac{41487}{67111} >$

Related Functions: Saturation.

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.