Gröbner Basen händisch
system:sage


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

{{{id=2|
f1=2*x^2*y+3*x+4*y
f2=3*x*y^2+4*x+5*y
///
}}}

{{{id=9|
f1
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}2 x^{2} y + 3 x + 4 y</span></html>
}}}

{{{id=10|
f2
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}3 x y^{2} + 4 x + 5 y</span></html>
}}}

{{{id=11|
f3=3*y*f1-2*x*f2;f3
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}-8 x^{2} - x y + 12 y^{2}</span></html>
}}}

{{{id=12|
f4=f1*4+y*f3;f4
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}- x y^{2} + 12 x + 12 y^{3} + 16 y</span></html>
}}}

{{{id=23|
f4a=f4*3+f2;f4a
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}40 x + 36 y^{3} + 53 y</span></html>
}}}

{{{id=24|
f5=f4a*x+5*f3;f5
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}36 x y^{3} + 48 x y + 60 y^{2}</span></html>
}}}

{{{id=25|
f5a=f5-y^3*36/40*f4a; f5a
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}48 x y - \frac{162}{5} y^{6} - \frac{477}{10} y^{4} + 60 y^{2}</span></html>
}}}

{{{id=26|
f5b=f5a-48/40*y*f4a;f5b
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{162}{5} y^{6} - \frac{909}{10} y^{4} - \frac{18}{5} y^{2}</span></html>
}}}

{{{id=27|
usw
///
}}}

{{{id=28|
Id=(f1,f2)*R; Id
///
Ideal (2*x^2*y + 3*x + 4*y, 3*x*y^2 + 4*x + 5*y) of Multivariate Polynomial Ring in x, y over Rational Field
}}}

{{{id=29|
Id.groebner_basis()
///
[x + 9/10*y^3 + 53/40*y, y^5 + 101/36*y^3 + 1/9*y]
}}}

{{{id=20|

///
}}}

{{{id=21|

///
}}}

{{{id=22|

///
}}}