Ideal-Membership-händisch
system:sage


{{{id=1|
R.<x,y>=PolynomialRing(QQ,2,order='invlex')
///
}}}

{{{id=2|
f1=2*x^2*y+3*x+4*y
f2=3*x*y^2+4*x+5*y
f3=40*x+53*y+36*y^3
f4=12*y+303*y^3+108*y^5
g=-8*x^2-x*y+12*y^2
///
}}}

{{{id=10|
f1, f2, f3, f4, g
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 x^{2} y + 4 y + 3 x, 3 x y^{2} + 5 y + 4 x, 36 y^{3} + 53 y + 40 x, 108 y^{5} + 303 y^{3} + 12 y, 12 y^{2} - x y - 8 x^{2}\right)</span></html>
}}}

{{{id=12|
g1=g+1/5*x*f3;g1
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\frac{36}{5} x y^{3} + 12 y^{2} + \frac{48}{5} x y</span></html>
}}}

{{{id=17|
g2a=g1-36/5*f2*y/3;g2a
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}0</span></html>
}}}

{{{id=13|
g2=g1-36/5*y^3/40*f3;g2
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{162}{25} y^{6} - \frac{477}{50} y^{4} + 12 y^{2} + \frac{48}{5} x y</span></html>
}}}

{{{id=14|
g3=g2-48/5*y*f3/40;g3
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{162}{25} y^{6} - \frac{909}{50} y^{4} - \frac{18}{25} y^{2}</span></html>
}}}

{{{id=15|
g4=g3+162/25*y*f4/108;g4
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}0</span></html>
}}}

{{{id=16|

///
}}}