Syzygies

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\centerline{\fbox{\parbox{68mm}{\Large\bf UserFunction Syzygies} }}
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\vspace{5mm}\noindent{\bf JanetBase}$(u:LDFMOD(C,zs,xs,O))$
The single argument $u$ has the type $LDFMOD$ which abbreviates
{\em LinearDifferentialFormModule}, it represents a system of linear
homogeneous partial differential equations. The meaning of its parameters is
as follows.

$C$: Coefficients.

$zs=\{z_1,z_2,\ldots \}$: Dependent variables in decreasing order.

$xs=\{x_1,x_2,\ldots \}$: Independent variables in decreasing order.

$O:LEX|GRLEX$: Term ordering.

\vspace*{1mm}\noindent{\bf Specification.}
Generators for the syzygy module are returned.

\vspace*{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The
output for each minisession is returned in a separate window.

\4{\tt l:=\{Df(z,x,2)-y/(x**2+x*y)*Df(z,x), Df(z,x,y)+1/(x+y)*Df(z,y),}

\hspace*{20mm}{\tt Df(z,y,2)+1/(x+y)*Df(z,y)\}; }

\5{\tt s:=Syzygies(l|LDFMOD(RATF Q,\{z\},\{x,y\},GRLEX)|);}

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The given module:
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$z_{x,x}-\fracsm{y}{x^{2}+xy}z_{y}+\fracsm{1}{x}z_{x}=e_1$
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$z_{x,y}+\fracsm{1}{x+y}z_{y}=e_2$
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$z_{y,y}+\fracsm{1}{x+y}z_{y}=e_3$


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The syzygies module:

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$<e_{2,y}-e_{3,x}+\fracsm{1}{x+y}e_2-\fracsm{1}{x+y}e_3,\hspace*{1mm}
  e_{1,y}-e_{2,x}-\fracsm{y}{x^{2}+xy}e_2+\fracsm{y}{x^{2}+xy}e_3>$ }}


\vspace*{1mm}\noindent{\bf Related Functions:}


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{\bf References}

F.~Schwarz, {\em Algorithmic Lie Theory for Solving Ordinary Differential Equations}, CRC Press, 2007.


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