#### UserFunction Product

Product(u,v:DPOLID(C,v,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. Generators for the product ideal of $u$ and $v$ are returned. In general they are not a Groebner basis.

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

I:={x*y+2*x-3,x**2+3*x+y};

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

Product(I,J|DPOLID(RATF Q,{x,y},GRLEX)|);

The given ideals:
$<xy+2x-3,x^{2}+3x+y>,$
$<3xy+y^{2}-x-1,x+3y-1>$
The product ideal:
$<3x^2y^2+5x^2y-2x^2+xy^3+2xy^2-10xy+x-3y^2+3,$
$x^2y+2x^2+3xy^2+5xy-5x-9y+3,$
$3x^3y-x^3+x^2y^2+9x^2y-4x^2+6xy^2-xy-3x+y^3-y,$
$x^3+3x^2y+2x^2+10xy-3x+3y^2-y>$

Related Functions. Sum, Intersection.

References.

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

D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms and 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 and , Springer, 2000.