{{{id=1| R.=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 /// \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) }}} {{{id=12| g1=g+1/5*x*f3;g1 /// \newcommand{\Bold}[1]{\mathbf{#1}}\frac{36}{5} x y^{3} + 12 y^{2} + \frac{48}{5} x y }}} {{{id=17| g2a=g1-36/5*f2*y/3;g2a /// \newcommand{\Bold}[1]{\mathbf{#1}}0 }}} {{{id=13| g2=g1-36/5*y^3/40*f3;g2 /// \newcommand{\Bold}[1]{\mathbf{#1}}-\frac{162}{25} y^{6} - \frac{477}{50} y^{4} + 12 y^{2} + \frac{48}{5} x y }}} {{{id=14| g3=g2-48/5*y*f3/40;g3 /// \newcommand{\Bold}[1]{\mathbf{#1}}-\frac{162}{25} y^{6} - \frac{909}{50} y^{4} - \frac{18}{25} y^{2} }}} {{{id=15| g4=g3+162/25*y*f4/108;g4 /// \newcommand{\Bold}[1]{\mathbf{#1}}0 }}} {{{id=16| /// }}}