\begin{patch}{RegularTriangularSetXmpPagePatch1}
\begin{paste}{RegularTriangularSetXmpPageFull1}{RegularTriangularSetXmpPageEmpty1}
\pastebutton{RegularTriangularSetXmpPageFull1}{\hidepaste}
\tab{5}\spadcommand{R := Integer\bound{R }}
\indentrel{3}\begin{verbatim}
   (1)  Integer
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty1}
\begin{paste}{RegularTriangularSetXmpPageEmpty1}{RegularTriangularSetXmpPagePatch1}
\pastebutton{RegularTriangularSetXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{R := Integer\bound{R }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch2}
\begin{paste}{RegularTriangularSetXmpPageFull2}{RegularTriangularSetXmpPageEmpty2}
\pastebutton{RegularTriangularSetXmpPageFull2}{\hidepaste}
\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
\indentrel{3}\begin{verbatim}
   (2)  [x,y,z,t]
                                     Type: List(Symbol)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty2}
\begin{paste}{RegularTriangularSetXmpPageEmpty2}{RegularTriangularSetXmpPagePatch2}
\pastebutton{RegularTriangularSetXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch3}
\begin{paste}{RegularTriangularSetXmpPageFull3}{RegularTriangularSetXmpPageEmpty3}
\pastebutton{RegularTriangularSetXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
\indentrel{3}\begin{verbatim}
   (3)  OrderedVariableList([x,y,z,t])
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty3}
\begin{paste}{RegularTriangularSetXmpPageEmpty3}{RegularTriangularSetXmpPagePatch3}
\pastebutton{RegularTriangularSetXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch4}
\begin{paste}{RegularTriangularSetXmpPageFull4}{RegularTriangularSetXmpPageEmpty4}
\pastebutton{RegularTriangularSetXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
\indentrel{3}\begin{verbatim}
   (4)  IndexedExponents(OrderedVariableList([x,y,z,t]))
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty4}
\begin{paste}{RegularTriangularSetXmpPageEmpty4}{RegularTriangularSetXmpPagePatch4}
\pastebutton{RegularTriangularSetXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch5}
\begin{paste}{RegularTriangularSetXmpPageFull5}{RegularTriangularSetXmpPageEmpty5}
\pastebutton{RegularTriangularSetXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
\indentrel{3}\begin{verbatim}
   (5)
  NewSparseMultivariatePolynomial(Integer,OrderedVariable
  List([x,y,z,t]))
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty5}
\begin{paste}{RegularTriangularSetXmpPageEmpty5}{RegularTriangularSetXmpPagePatch5}
\pastebutton{RegularTriangularSetXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch6}
\begin{paste}{RegularTriangularSetXmpPageFull6}{RegularTriangularSetXmpPageEmpty6}
\pastebutton{RegularTriangularSetXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
\indentrel{3}\begin{verbatim}
   (6)  x
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty6}
\begin{paste}{RegularTriangularSetXmpPageEmpty6}{RegularTriangularSetXmpPagePatch6}
\pastebutton{RegularTriangularSetXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch7}
\begin{paste}{RegularTriangularSetXmpPageFull7}{RegularTriangularSetXmpPageEmpty7}
\pastebutton{RegularTriangularSetXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
\indentrel{3}\begin{verbatim}
   (7)  y
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty7}
\begin{paste}{RegularTriangularSetXmpPageEmpty7}{RegularTriangularSetXmpPagePatch7}
\pastebutton{RegularTriangularSetXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch8}
\begin{paste}{RegularTriangularSetXmpPageFull8}{RegularTriangularSetXmpPageEmpty8}
\pastebutton{RegularTriangularSetXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
\indentrel{3}\begin{verbatim}
   (8)  z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty8}
\begin{paste}{RegularTriangularSetXmpPageEmpty8}{RegularTriangularSetXmpPagePatch8}
\pastebutton{RegularTriangularSetXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch9}
\begin{paste}{RegularTriangularSetXmpPageFull9}{RegularTriangularSetXmpPageEmpty9}
\pastebutton{RegularTriangularSetXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
\indentrel{3}\begin{verbatim}
   (9)  t
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty9}
\begin{paste}{RegularTriangularSetXmpPageEmpty9}{RegularTriangularSetXmpPagePatch9}
\pastebutton{RegularTriangularSetXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch10}
\begin{paste}{RegularTriangularSetXmpPageFull10}{RegularTriangularSetXmpPageEmpty10}
\pastebutton{RegularTriangularSetXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
\indentrel{3}\begin{verbatim}
   (10)
  RegularTriangularSet(Integer,IndexedExponents(OrderedVa
  riableList([x,y,z,t])),OrderedVariableList([x,y,z,t]),N
  ewSparseMultivariatePolynomial(Integer,OrderedVariableL
  ist([x,y,z,t])))
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty10}
\begin{paste}{RegularTriangularSetXmpPageEmpty10}{RegularTriangularSetXmpPagePatch10}
\pastebutton{RegularTriangularSetXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch11}
\begin{paste}{RegularTriangularSetXmpPageFull11}{RegularTriangularSetXmpPageEmpty11}
\pastebutton{RegularTriangularSetXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{p1 := x ^ 31 - x ^ 6 - x - y\free{x }\free{y }\bound{p1 }}
\indentrel{3}\begin{verbatim}
          31    6
   (11)  x   - x  - x - y
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty11}
\begin{paste}{RegularTriangularSetXmpPageEmpty11}{RegularTriangularSetXmpPagePatch11}
\pastebutton{RegularTriangularSetXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{p1 := x ^ 31 - x ^ 6 - x - y\free{x }\free{y }\bound{p1 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch12}
\begin{paste}{RegularTriangularSetXmpPageFull12}{RegularTriangularSetXmpPageEmpty12}
\pastebutton{RegularTriangularSetXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{p2 := x ^ 8 - z\free{x }\free{z }\bound{p2 }}
\indentrel{3}\begin{verbatim}
          8
   (12)  x  - z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty12}
\begin{paste}{RegularTriangularSetXmpPageEmpty12}{RegularTriangularSetXmpPagePatch12}
\pastebutton{RegularTriangularSetXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{p2 := x ^ 8 - z\free{x }\free{z }\bound{p2 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch13}
\begin{paste}{RegularTriangularSetXmpPageFull13}{RegularTriangularSetXmpPageEmpty13}
\pastebutton{RegularTriangularSetXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{p3 := x ^ 10 - t\free{x }\free{t }\bound{p3 }}
\indentrel{3}\begin{verbatim}
          10
   (13)  x   - t
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty13}
\begin{paste}{RegularTriangularSetXmpPageEmpty13}{RegularTriangularSetXmpPagePatch13}
\pastebutton{RegularTriangularSetXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{p3 := x ^ 10 - t\free{x }\free{t }\bound{p3 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch14}
\begin{paste}{RegularTriangularSetXmpPageFull14}{RegularTriangularSetXmpPageEmpty14}
\pastebutton{RegularTriangularSetXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
\indentrel{3}\begin{verbatim}
           31    6          8      10
   (14)  [x   - x  - x - y,x  - z,x   - t]
Type: List(NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t])))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty14}
\begin{paste}{RegularTriangularSetXmpPageEmpty14}{RegularTriangularSetXmpPagePatch14}
\pastebutton{RegularTriangularSetXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch15}
\begin{paste}{RegularTriangularSetXmpPageFull15}{RegularTriangularSetXmpPageEmpty15}
\pastebutton{RegularTriangularSetXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
\indentrel{3}\begin{verbatim}
   (15)
   [
       5    4       2     3     8     5    3    2
     {z  - t , t z y  + 2z y - t  + 2t  + t  - t ,
        4                2
      (t  - t)x - t y - z }
     ]
Type: List(RegularTriangularSet(Integer,IndexedExponents(OrderedVariableList([x,y,z,t])),OrderedVariableList([x,y,z,t]),NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty15}
\begin{paste}{RegularTriangularSetXmpPageEmpty15}{RegularTriangularSetXmpPagePatch15}
\pastebutton{RegularTriangularSetXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch16}
\begin{paste}{RegularTriangularSetXmpPageFull16}{RegularTriangularSetXmpPageEmpty16}
\pastebutton{RegularTriangularSetXmpPageFull16}{\hidepaste}
\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
\indentrel{3}\begin{verbatim}
   (16)
   [
       5    4       2     3     8     5    3    2
     {z  - t , t z y  + 2z y - t  + 2t  + t  - t ,
        4                2
      (t  - t)x - t y - z }
     ,
      3      5          2     3         2
    {t  - 1,z  - t,t z y  + 2z y + 1,z x  - t},
    {t,z,y,x}]
Type: List(RegularTriangularSet(Integer,IndexedExponents(OrderedVariableList([x,y,z,t])),OrderedVariableList([x,y,z,t]),NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty16}
\begin{paste}{RegularTriangularSetXmpPageEmpty16}{RegularTriangularSetXmpPagePatch16}
\pastebutton{RegularTriangularSetXmpPageEmpty16}{\showpaste}
\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch17}
\begin{paste}{RegularTriangularSetXmpPageFull17}{RegularTriangularSetXmpPageEmpty17}
\pastebutton{RegularTriangularSetXmpPageFull17}{\hidepaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
\indentrel{3}\begin{verbatim}
   (17)  [1,0,0]
                         Type: List(NonNegativeInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty17}
\begin{paste}{RegularTriangularSetXmpPageEmpty17}{RegularTriangularSetXmpPagePatch17}
\pastebutton{RegularTriangularSetXmpPageEmpty17}{\showpaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch18}
\begin{paste}{RegularTriangularSetXmpPageFull18}{RegularTriangularSetXmpPageEmpty18}
\pastebutton{RegularTriangularSetXmpPageFull18}{\hidepaste}
\tab{5}\spadcommand{f1 := y^2*z+2*x*y*t-2*x-z\free{z }\free{x }\free{y }\free{t }\bound{f1 }}
\indentrel{3}\begin{verbatim}
                          2
   (18)  (2t y - 2)x + z y  - z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty18}
\begin{paste}{RegularTriangularSetXmpPageEmpty18}{RegularTriangularSetXmpPagePatch18}
\pastebutton{RegularTriangularSetXmpPageEmpty18}{\showpaste}
\tab{5}\spadcommand{f1 := y^2*z+2*x*y*t-2*x-z\free{z }\free{x }\free{y }\free{t }\bound{f1 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch19}
\begin{paste}{RegularTriangularSetXmpPageFull19}{RegularTriangularSetXmpPageEmpty19}
\pastebutton{RegularTriangularSetXmpPageFull19}{\hidepaste}
\tab{5}\spadcommand{f2 := -x^3*z+ 4*x*y^2*z+ 4*x^2*y*t+ 2*y^3*t+ 4*x^2- 10*y^2+ 4*x*z- 10*y*t+ 2\free{z }\free{x }\free{y }\free{t }\bound{f2 }}
\indentrel{3}\begin{verbatim}
   (19)
          3              2        2              3      2
     - z x  + (4t y + 4)x  + (4z y  + 4z)x + 2t y  - 10y
   + 
     - 10t y + 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty19}
\begin{paste}{RegularTriangularSetXmpPageEmpty19}{RegularTriangularSetXmpPagePatch19}
\pastebutton{RegularTriangularSetXmpPageEmpty19}{\showpaste}
\tab{5}\spadcommand{f2 := -x^3*z+ 4*x*y^2*z+ 4*x^2*y*t+ 2*y^3*t+ 4*x^2- 10*y^2+ 4*x*z- 10*y*t+ 2\free{z }\free{x }\free{y }\free{t }\bound{f2 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch20}
\begin{paste}{RegularTriangularSetXmpPageFull20}{RegularTriangularSetXmpPageEmpty20}
\pastebutton{RegularTriangularSetXmpPageFull20}{\hidepaste}
\tab{5}\spadcommand{f3 := 2*y*z*t+x*t^2-x-2*z\free{z }\free{x }\free{y }\free{t }\bound{f3 }}
\indentrel{3}\begin{verbatim}
           2
   (20)  (t  - 1)x + 2t z y - 2z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty20}
\begin{paste}{RegularTriangularSetXmpPageEmpty20}{RegularTriangularSetXmpPagePatch20}
\pastebutton{RegularTriangularSetXmpPageEmpty20}{\showpaste}
\tab{5}\spadcommand{f3 := 2*y*z*t+x*t^2-x-2*z\free{z }\free{x }\free{y }\free{t }\bound{f3 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch21}
\begin{paste}{RegularTriangularSetXmpPageFull21}{RegularTriangularSetXmpPageEmpty21}
\pastebutton{RegularTriangularSetXmpPageFull21}{\hidepaste}
\tab{5}\spadcommand{f4 := -x*z^3+ 4*y*z^2*t+ 4*x*z*t^2+ 2*y*t^3+ 4*x*z+ 4*z^2-10*y*t- 10*t^2+2\free{z }\free{x }\free{y }\free{t }\bound{f4 }}
\indentrel{3}\begin{verbatim}
   (21)
         3      2                2     3             2
     (- z  + (4t  + 4)z)x + (4t z  + 2t  - 10t)y + 4z
   + 
          2
     - 10t  + 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty21}
\begin{paste}{RegularTriangularSetXmpPageEmpty21}{RegularTriangularSetXmpPagePatch21}
\pastebutton{RegularTriangularSetXmpPageEmpty21}{\showpaste}
\tab{5}\spadcommand{f4 := -x*z^3+ 4*y*z^2*t+ 4*x*z*t^2+ 2*y*t^3+ 4*x*z+ 4*z^2-10*y*t- 10*t^2+2\free{z }\free{x }\free{y }\free{t }\bound{f4 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch22}
\begin{paste}{RegularTriangularSetXmpPageFull22}{RegularTriangularSetXmpPageEmpty22}
\pastebutton{RegularTriangularSetXmpPageFull22}{\hidepaste}
\tab{5}\spadcommand{lf := [f1, f2, f3, f4]\free{f1 }\free{f2 }\free{f3 }\free{f4 }\bound{lf }}
\indentrel{3}\begin{verbatim}
   (22)
                     2
   [(2t y - 2)x + z y  - z,

            3              2        2              3
       - z x  + (4t y + 4)x  + (4z y  + 4z)x + 2t y
     + 
            2
       - 10y  - 10t y + 2
     ,
      2
    (t  - 1)x + 2t z y - 2z,

           3      2                2     3             2
       (- z  + (4t  + 4)z)x + (4t z  + 2t  - 10t)y + 4z
     + 
            2
       - 10t  + 2
     ]
Type: List(NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t])))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty22}
\begin{paste}{RegularTriangularSetXmpPageEmpty22}{RegularTriangularSetXmpPagePatch22}
\pastebutton{RegularTriangularSetXmpPageEmpty22}{\showpaste}
\tab{5}\spadcommand{lf := [f1, f2, f3, f4]\free{f1 }\free{f2 }\free{f3 }\free{f4 }\bound{lf }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch23}
\begin{paste}{RegularTriangularSetXmpPageFull23}{RegularTriangularSetXmpPageEmpty23}
\pastebutton{RegularTriangularSetXmpPageFull23}{\hidepaste}
\tab{5}\spadcommand{zeroSetSplit(lf)$T\free{lf }\free{T }}
\indentrel{3}\begin{verbatim}
   (23)
   [
       2       8      6       2
     {t  - 1, z  - 16z  + 256z  - 256, t y - 1,
        3            2
      (z  - 8z)x - 8z  + 16}
     ,
       2      2     2
    {3t  + 1,z  - 7t  - 1,y + t,x + z},
      8      6      2         3            2
    {t  - 10t  + 10t  - 1,z,(t  - 5t)y - 5t  + 1,x},
      2      2
    {t  + 3,z  - 4,y + t,x - z}]
Type: List(RegularTriangularSet(Integer,IndexedExponents(OrderedVariableList([x,y,z,t])),OrderedVariableList([x,y,z,t]),NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty23}
\begin{paste}{RegularTriangularSetXmpPageEmpty23}{RegularTriangularSetXmpPagePatch23}
\pastebutton{RegularTriangularSetXmpPageEmpty23}{\showpaste}
\tab{5}\spadcommand{zeroSetSplit(lf)$T\free{lf }\free{T }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch24}
\begin{paste}{RegularTriangularSetXmpPageFull24}{RegularTriangularSetXmpPageEmpty24}
\pastebutton{RegularTriangularSetXmpPageFull24}{\hidepaste}
\tab{5}\spadcommand{lts2 := zeroSetSplit(lf,false)$T\free{lf }\free{T }\bound{lts2 }}
\indentrel{3}\begin{verbatim}
   (24)
      8      6      2         3            2
   [{t  - 10t  + 10t  - 1,z,(t  - 5t)y - 5t  + 1,x},

       2       8      6       2
     {t  - 1, z  - 16z  + 256z  - 256, t y - 1,
        3            2
      (z  - 8z)x - 8z  + 16}
     ,
       2      2     2
    {3t  + 1,z  - 7t  - 1,y + t,x + z},
      2      2
    {t  + 3,z  - 4,y + t,x - z}]
Type: List(RegularTriangularSet(Integer,IndexedExponents(OrderedVariableList([x,y,z,t])),OrderedVariableList([x,y,z,t]),NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty24}
\begin{paste}{RegularTriangularSetXmpPageEmpty24}{RegularTriangularSetXmpPagePatch24}
\pastebutton{RegularTriangularSetXmpPageEmpty24}{\showpaste}
\tab{5}\spadcommand{lts2 := zeroSetSplit(lf,false)$T\free{lf }\free{T }\bound{lts2 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch25}
\begin{paste}{RegularTriangularSetXmpPageFull25}{RegularTriangularSetXmpPageEmpty25}
\pastebutton{RegularTriangularSetXmpPageFull25}{\hidepaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts2]\free{lts2 }}
\indentrel{3}\begin{verbatim}
   (25)  [0,0,0,0]
                         Type: List(NonNegativeInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty25}
\begin{paste}{RegularTriangularSetXmpPageEmpty25}{RegularTriangularSetXmpPagePatch25}
\pastebutton{RegularTriangularSetXmpPageEmpty25}{\showpaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts2]\free{lts2 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch26}
\begin{paste}{RegularTriangularSetXmpPageFull26}{RegularTriangularSetXmpPageEmpty26}
\pastebutton{RegularTriangularSetXmpPageFull26}{\hidepaste}
\tab{5}\spadcommand{degrees := [degree(ts) for ts in lts2]\free{lts2 }\bound{degrees }}
\indentrel{3}\begin{verbatim}
   (26)  [8,16,4,4]
                         Type: List(NonNegativeInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty26}
\begin{paste}{RegularTriangularSetXmpPageEmpty26}{RegularTriangularSetXmpPagePatch26}
\pastebutton{RegularTriangularSetXmpPageEmpty26}{\showpaste}
\tab{5}\spadcommand{degrees := [degree(ts) for ts in lts2]\free{lts2 }\bound{degrees }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch27}
\begin{paste}{RegularTriangularSetXmpPageFull27}{RegularTriangularSetXmpPageEmpty27}
\pastebutton{RegularTriangularSetXmpPageFull27}{\hidepaste}
\tab{5}\spadcommand{reduce(+,degrees)\free{degrees }}
\indentrel{3}\begin{verbatim}
   (27)  32
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty27}
\begin{paste}{RegularTriangularSetXmpPageEmpty27}{RegularTriangularSetXmpPagePatch27}
\pastebutton{RegularTriangularSetXmpPageEmpty27}{\showpaste}
\tab{5}\spadcommand{reduce(+,degrees)\free{degrees }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch28}
\begin{paste}{RegularTriangularSetXmpPageFull28}{RegularTriangularSetXmpPageEmpty28}
\pastebutton{RegularTriangularSetXmpPageFull28}{\hidepaste}
\tab{5}\spadcommand{u : R := 2\free{R }\bound{u }}
\indentrel{3}\begin{verbatim}
   (28)  2
                                          Type: Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty28}
\begin{paste}{RegularTriangularSetXmpPageEmpty28}{RegularTriangularSetXmpPagePatch28}
\pastebutton{RegularTriangularSetXmpPageEmpty28}{\showpaste}
\tab{5}\spadcommand{u : R := 2\free{R }\bound{u }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch29}
\begin{paste}{RegularTriangularSetXmpPageFull29}{RegularTriangularSetXmpPageEmpty29}
\pastebutton{RegularTriangularSetXmpPageFull29}{\hidepaste}
\tab{5}\spadcommand{q1 := 2*(u-1)^2+ 2*(x-z*x+z^2)+ y^2*(x-1)^2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u^2*t^2*(1-2*z)+ 2*u*t^2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u^2-2*u)*z^2*t^2+ 2*u^2*z^2+ 4*u*(1-u)*z+ t^2*(z-x)^2\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q1 }}
\indentrel{3}\begin{verbatim}
   (29)
       2           2  2
     (y  - 2t y + t )x
   + 
          2                            2           2
     (- 2y  + ((2t + 4)z + 2t)y + (- 2t  + 2)z - 4t  - 2)x
   + 
      2                      2       2          2
     y  + (- 2t z - 4t)y + (t  + 10)z  - 8z + 4t  + 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty29}
\begin{paste}{RegularTriangularSetXmpPageEmpty29}{RegularTriangularSetXmpPagePatch29}
\pastebutton{RegularTriangularSetXmpPageEmpty29}{\showpaste}
\tab{5}\spadcommand{q1 := 2*(u-1)^2+ 2*(x-z*x+z^2)+ y^2*(x-1)^2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u^2*t^2*(1-2*z)+ 2*u*t^2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u^2-2*u)*z^2*t^2+ 2*u^2*z^2+ 4*u*(1-u)*z+ t^2*(z-x)^2\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q1 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch30}
\begin{paste}{RegularTriangularSetXmpPageFull30}{RegularTriangularSetXmpPageEmpty30}
\pastebutton{RegularTriangularSetXmpPageFull30}{\hidepaste}
\tab{5}\spadcommand{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z^2*t\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q2 }}
\indentrel{3}\begin{verbatim}
                                               2
   (30)  (- 3z y + 2t z + t)x + (z + 4)y + 4t z  - 7t z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty30}
\begin{paste}{RegularTriangularSetXmpPageEmpty30}{RegularTriangularSetXmpPagePatch30}
\pastebutton{RegularTriangularSetXmpPageEmpty30}{\showpaste}
\tab{5}\spadcommand{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z^2*t\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q2 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch31}
\begin{paste}{RegularTriangularSetXmpPageFull31}{RegularTriangularSetXmpPageEmpty31}
\pastebutton{RegularTriangularSetXmpPageFull31}{\hidepaste}
\tab{5}\spadcommand{q3 := -u^2*(z-1)^2+ 2*z*(z-x)-2*(x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q3 }}
\indentrel{3}\begin{verbatim}
                         2
   (31)  (- 2z - 2)x - 2z  + 8z - 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty31}
\begin{paste}{RegularTriangularSetXmpPageEmpty31}{RegularTriangularSetXmpPagePatch31}
\pastebutton{RegularTriangularSetXmpPageEmpty31}{\showpaste}
\tab{5}\spadcommand{q3 := -u^2*(z-1)^2+ 2*z*(z-x)-2*(x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q3 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch32}
\begin{paste}{RegularTriangularSetXmpPageFull32}{RegularTriangularSetXmpPageEmpty32}
\pastebutton{RegularTriangularSetXmpPageFull32}{\hidepaste}
\tab{5}\spadcommand{q4 := u^2+4*(z-x^2)+3*y^2*(x-1)^2- 3*t^2*(z-x)^2 +3*u^2*t^2*(z-1)^2+u^2*z*(z-2)+6*u*t*y*(z+x+z*x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q4 }}
\indentrel{3}\begin{verbatim}
   (32)
        2     2      2        2                      2
     (3y  - 3t  - 4)x  + (- 6y  + (12t z + 12t)y + 6t z)x
   + 
       2                       2      2         2
     3y  + (12t z - 12t)y + (9t  + 4)z  + (- 24t  - 4)z
   + 
        2
     12t  + 4
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty32}
\begin{paste}{RegularTriangularSetXmpPageEmpty32}{RegularTriangularSetXmpPagePatch32}
\pastebutton{RegularTriangularSetXmpPageEmpty32}{\showpaste}
\tab{5}\spadcommand{q4 := u^2+4*(z-x^2)+3*y^2*(x-1)^2- 3*t^2*(z-x)^2 +3*u^2*t^2*(z-1)^2+u^2*z*(z-2)+6*u*t*y*(z+x+z*x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q4 }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch33}
\begin{paste}{RegularTriangularSetXmpPageFull33}{RegularTriangularSetXmpPageEmpty33}
\pastebutton{RegularTriangularSetXmpPageFull33}{\hidepaste}
\tab{5}\spadcommand{lq := [q1, q2, q3, q4]\free{q1 }\free{q2 }\free{q3 }\free{q4 }\bound{lq }}
\indentrel{3}\begin{verbatim}
   (33)
   [
         2           2  2
       (y  - 2t y + t )x
     + 
               2                            2           2
           - 2y  + ((2t + 4)z + 2t)y + (- 2t  + 2)z - 4t
         + 
           - 2
      *
         x
     + 
        2                      2       2          2
       y  + (- 2t z - 4t)y + (t  + 10)z  - 8z + 4t  + 2
     ,
                                          2
    (- 3z y + 2t z + t)x + (z + 4)y + 4t z  - 7t z,
                    2
    (- 2z - 2)x - 2z  + 8z - 2,

          2     2      2
       (3y  - 3t  - 4)x
     + 
            2                      2        2
       (- 6y  + (12t z + 12t)y + 6t z)x + 3y
     + 
                           2      2         2            2
       (12t z - 12t)y + (9t  + 4)z  + (- 24t  - 4)z + 12t
     + 
       4
     ]
Type: List(NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t])))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty33}
\begin{paste}{RegularTriangularSetXmpPageEmpty33}{RegularTriangularSetXmpPagePatch33}
\pastebutton{RegularTriangularSetXmpPageEmpty33}{\showpaste}
\tab{5}\spadcommand{lq := [q1, q2, q3, q4]\free{q1 }\free{q2 }\free{q3 }\free{q4 }\bound{lq }}
\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPagePatch34}
\begin{paste}{RegularTriangularSetXmpPageFull34}{RegularTriangularSetXmpPageEmpty34}
\pastebutton{RegularTriangularSetXmpPageFull34}{\hidepaste}
\tab{5}\spadcommand{zeroSetSplit(lq,true,true)$T\free{lq }\free{T }}
\indentrel{3}\begin{verbatim}

   *** QCMPACK Statistics ***
      Table     size:  36
      Entries reused:  255

   *** REGSETGCD: Gcd Statistics ***
      Table     size:  125
      Entries reused:  0

   *** REGSETGCD: Inv Set Statistics ***
      Table     size:  30
      Entries reused:  0
   (34)
   [
     {
                         24                   23
         960725655771966t   + 386820897948702t
       + 
                          22                    21
         8906817198608181t   + 2704966893949428t
       + 
                           20                    19
         37304033340228264t   + 7924782817170207t
       + 
                           18                     17
         93126799040354990t   + 13101273653130910t
       + 
                            16                     15
         156146250424711858t   + 16626490957259119t
       + 
                            14                     13
         190699288479805763t   + 24339173367625275t
       + 
                            12                     11
         180532313014960135t   + 35288089030975378t
       + 
                            10                     9
         135054975747656285t   + 34733736952488540t
       + 
                           8                     7
         75947600354493972t  + 19772555692457088t
       + 
                           6                    5
         28871558573755428t  + 5576152439081664t
       + 
                          4                   3
         6321711820352976t  + 438314209312320t
       + 
                       2
       581105748367008t  - 60254467992576t + 1449115951104
       ,

              266042108694913023855152657370520823616684_
               74181372891857784
            *
                23
               t
           + 
              443104378424686086067294899528296664238693_
               556855017735265295
            *
                22
               t
           + 
              279078393286701234679141342358988327155321_
               305829547090310242
            *
                21
               t
           + 
              339027636141323246510761717661554305462062_
               6391823613392185226
            *
                20
               t
           + 
              941478179503540575554198645220352803719793_
               196473813837434129
            *
                19
               t
           + 
              115478551946794752422116967496739493525857_
               47674184320988144390
            *
                18
               t
           + 
              134360956676559778988170165669941321646721_
               5660333356417241432
            *
                17
               t
           + 
              232338138681478735039335516171756408598991_
               02987800663566699334
            *
                16
               t
           + 
              869574020537672336950845440508790740850931_
               336484983573386433
            *
                15
               t
           + 
              315615543058769348754194614869699265542417_
               50065103460820476969
            *
                14
               t
           + 
              127140099028771748744206595254773187955482_
               3889855386072264931
            *
                13
               t
           + 
              319450899138637360448025269640795401983370_
               49550503295825160523
            *
                12
               t
           + 
              373873570428814450987137156023284588443910_
               2270778010470931960
            *
                11
               t
           + 
              252939975123914120261446014357711315875619_
               05532992045692885927
            *
                10
               t
           + 
              521023900984606712346926279987005277341047_
               1135950175008046524
            *
                9
               t
           + 
              150838879869302971662598705686082704274031_
               87606238713491129188
            *
                8
               t
           + 
              352208723469293012638368627077577955348176_
               9125670839075109000
            *
                7
               t
           + 
              607994520039568101308653379256888649110124_
               4247440034969288588
            *
                6
               t
           + 
              109063485243390088819991375624798602319698_
               7723469934933603680
            *
                5
               t
           + 
              140581943087190710229443253753833540210283_
               8994019667487458352
            *
                4
               t
           + 
              880715279503204500725366712655077488783478_
               28884933605202432
            *
                3
               t
           + 
              135882489433640933229781177155977768016065_
               765482378657129440
            *
                2
               t
           + 
             -
                 139572834428822622305598946074003140825_
                  16690749975646520320
               *
                  t
           + 
            33463769297318929927725832570930847259211711_
             2855749713920
        *
           z
       + 
          8567175484043952879756725964506833932149637101_
           090521164936
        *
            23
           t
       + 
          1497923928642017918457083740327289424987975192_
           51667250945721
        *
            22
           t
       + 
          7725837178364582215741086158215976413812300307_
           4190374021550
        *
            21
           t
       + 
          1108862254126854214498918940708612211184560556_
           764334742191654
        *
            20
           t
       + 
          2132504944606788652197744801068260537838157896_
           21501732672327
        *
            19
           t
       + 
          3668929075160666195729177894178343514501987898_
           410131431699882
        *
            18
           t
       + 
          1713889064710018728794901243687482363147654590_
           39567820048872
        *
            17
           t
       + 
          7192430746914602166660233477331022483144921771_
           645523139658986
        *
            16
           t
       + 
         -
             1287986746896900728128799656330902919596631_
              43108437362453385
           *
               15
              t
       + 
          9553010858341425909306423132921134040856028790_
           803526430270671
        *
            14
           t
       + 
         -
             1329609624567549287453868764630043782465845_
              8709144441096603
           *
               13
              t
       + 
          9475806805814145326383085518325333106881690568_
           644274964864413
        *
            12
           t
       + 
          8032346879251334588616598556640849276062987947_
           99856265539336
        *
            11
           t
       + 
          7338202759292865165994622349207516400662174302_
           614595173333825
        *
            10
           t
       + 
          1308004628480367351164369613111971668880538855_
           640917200187108
        *
            9
           t
       + 
          4268059455741255498880229598973705747098216067_
           697754352634748
        *
            8
           t
       + 
          8928935268585140957913187759040933001030456015_
           14470613580600
        *
            7
           t
       + 
          1679152575460683956631925852181341501981598137_
           465328797013652
        *
            6
           t
       + 
          2697574157679229803789671541433578355441131582_
           80591408043936
        *
            5
           t
       + 
          3809515278646575290335808298012827240813453726_
           80202920198224
        *
            4
           t
       + 
          1978554529422849503299882693760134113272503533_
           9452913286656
        *
            3
           t
       + 
          3647741205738478294236663530339663776330392817_
           4935079178528
        *
            2
           t
       + 
         -
             3722212879279038648713080422224976273210890_
              229485838670848
           *
              t
       + 
        890797248531143483612306344840138620247285999068_
         74105856
       ,
         3      2                  3       2
      (3z  - 11z  + 8z + 4)y + 2t z  + 4t z  - 5t z - t,
                  2
      (z + 1)x + z  - 4z + 1}
     ]
Type: List(RegularTriangularSet(Integer,IndexedExponents(OrderedVariableList([x,y,z,t])),OrderedVariableList([x,y,z,t]),NewSparseMultivariatePolynomial(Integer,OrderedVariableList([x,y,z,t]))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{RegularTriangularSetXmpPageEmpty34}
\begin{paste}{RegularTriangularSetXmpPageEmpty34}{RegularTriangularSetXmpPagePatch34}
\pastebutton{RegularTriangularSetXmpPageEmpty34}{\showpaste}
\tab{5}\spadcommand{zeroSetSplit(lq,true,true)$T\free{lq }\free{T }}
\end{paste}\end{patch}

