diff --git a/changelog b/changelog index 9398dcc..056933f 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20080901 tpd src/input/Makefile add start of multivar poly tests +20080901 tpd src/input/bini.input add start of multivar poly tests 20080831 tpd src/algebra/algfact.spad graphviz dotfile decoration 20080831 tpd src/algebra/algext.spad graphviz dotfile decoration 20080830 tpd src/algebra/trigcat.spad graphviz dotfile decoration diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 8bc5aa4..93c2ee7 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -275,7 +275,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ arrows.regress asinatan.regress asinhatanh.regress \ assign.regress atansqrt.regress \ asec.regress bags.regress bbtree.regress besselk.regress \ - binary.regress biquat.regress \ + binary.regress bini.regress biquat.regress \ bop.regress bstree.regress bouquet.regress \ bug100.regress bug101.regress \ bug103.regress bug10069.regress \ @@ -520,8 +520,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/asec.input \ ${OUT}/bags.input ${OUT}/bbtree.input ${OUT}/besselk.input \ ${OUT}/bern.input \ - ${OUT}/bernpoly.input ${OUT}/binary.input ${OUT}/biquat.input \ - ${OUT}/bop.input \ + ${OUT}/bernpoly.input ${OUT}/binary.input ${OUT}/bini.input \ + ${OUT}/biquat.input ${OUT}/bop.input \ ${OUT}/bouquet.input ${OUT}/bstree.input ${OUT}/bug6357.input \ ${OUT}/bug9057.input ${OUT}/bug100.input ${OUT}/bug101.input \ ${OUT}/bug103.input \ @@ -729,7 +729,7 @@ DOCFILES= \ ${DOC}/bbtree.input.dvi ${DOC}/besselk.input.dvi \ ${DOC}/bern.input.dvi \ ${DOC}/bernpoly.input.dvi ${DOC}/binary.input.dvi \ - ${DOC}/biquat.input.dvi \ + ${DOC}/bini.input.dvi ${DOC}/biquat.input.dvi \ ${DOC}/bop.input.dvi ${DOC}/bouquet.input.dvi \ ${DOC}/bstree.input.dvi ${DOC}/bug10069.input.dvi \ ${DOC}/bug100.input.dvi ${DOC}/bug101.input.dvi \ diff --git a/src/input/bini.input.pamphlet b/src/input/bini.input.pamphlet new file mode 100644 index 0000000..eac6e13 --- /dev/null +++ b/src/input/bini.input.pamphlet @@ -0,0 +1,2922 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input bini.input} +\author{D. Bini and B. Mourrain} +\maketitle +\begin{abstract} +The POSSO multivariate polynomial test suite. +\end{abstract} +\eject +\tableofcontents +\eject +<<*>>= +)spool bini.output +)set message test on +)set message type off +)set message auto off +)clear all + +@ +Bini and Mourrain define a hierarchy of classes for polynomial solutions: +\begin{verbatim} + 1 Complete_Intersection + 1 Bezoutian_Matrix + 1 Table_of_Multiplication + 1 RUR + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 2 Hermite + 1 Real_Solve + 2 Count_the_real_solutions + 3 Eigenvector + 1 Solve + 2 Real_Solve + 2 Residue + 1 Table_of_Multiplication + 1 RUR + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 2 Hermite + 1 Real_Solve + 2 Count_the_real_solutions + 3 Eigenvector + 1 Solve + 2 Real_Solve + 2 Hermite + 1 Real_Solve + 2 Count_the_real_solutions + 3 RUR + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 4 Elimination_of_variables + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 2 General_System + 1 Grobner + 1 Table_of_Multiplication + 1 RUR + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 2 Hermite + 1 Real_Solve + 2 Count_the_real_solutions + 3 Eigenvector + 1 Solve + 2 Real_Solve + 2 Elimination_of_variables + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 3 Count_the_solutions + 3 Generic_Complete_Intersection + 1 Resultant_Matrix + 1 Table_of_Multiplication + 1 RUR + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 2 Hermite + 1 Real_Solve + 2 Count_the_real_solutions + 3 Eigenvector + 1 Solve + 2 Real_Solve + 2 Elimination_of_variables + 1 Uspensky + 1 Real_Solve + 2 Count_the_real_solutions + 2 Uni_Solve + 1 Solve + 2 Mix_Volume + 1 Count_the_solutions +\end{verbatim} + +\section{Bronstein2} +Paramaters $a,b$ and variables $x,y$) +<<*>>= +)clear all + +--S 1 +t1:=2*y^2*(y^2+x^2)+(b^2-3*a^2)*y^2-2*b*y^2*(x+y)+2*a^2*b*(y+x)_ + -a^2*x^2+a^2*(a^2-b^2) +--R +--R +--R (1) +--R 4 3 2 2 2 2 2 2 2 2 2 2 4 +--R 2y - 2b y + (2x - 2b x + b - 3a )y + 2a b y - a x + 2a b x - a b + a +--E 1 + +--S 2 +t2:=4*y^3+4*y*(y^2+x^2)-2*b*y^2-4*b*y*(y+x)+2*(b^2-3*a^2)*y+2*a^2*b +--R +--R +--R 3 2 2 2 2 2 +--R (2) 8y - 6b y + (4x - 4b x + 2b - 6a )y + 2a b +--E 2 + +--S 3 +t3:=4*x*y^2-2*b*y^2-2*a^2*x+2*a^2*b +--R +--R +--R 2 2 2 +--R (3) (4x - 2b)y - 2a x + 2a b +--E 3 + +@ +\section{CzaporGeddes1} +Variables $x,y,z$ +<<*>>= +)clear all + +--S 4 +t1:=8*x^2-2*x*y-6*x*z+3*x+3*y^2-7*y*z+10*y+10*z^2-8*z-4 +--R +--R +--R 2 2 2 +--R (1) 10z + (- 7y - 6x - 8)z + 3y + (- 2x + 10)y + 8x + 3x - 4 +--E 4 + +--S 5 +t2:=10*x^2-2*x*y+6*x*z-6*x+9*y^2-y*z-4*y-2*z^2+5*z-9 +--R +--R +--R 2 2 2 +--R (2) - 2z + (- y + 6x + 5)z + 9y + (- 2x - 4)y + 10x - 6x - 9 +--E 5 + +--S 6 +t3:=5*x^2+8*x*y+4*x*z+8*x+9*y^2-6*y*z+2*y-z^2-7*x+5 +--R +--R +--R 2 2 2 +--R (3) - z + (- 6y + 4x)z + 9y + (8x + 2)y + 5x + x + 5 +--E 6 + +@ +\section{CzaporGeddes2} +Parameter $b$ and variables $c,d,p,q$ +<<*>>= + +)clear all +--S 7 +t1:=2*(b-1)^2 + 2*(q-p*q+p^2) + c^2*(q-1)^2 -2*b*q + 2*c*d*(1-q)*(q-p)_ + +2*b*p*q*d*(d-c) + b^2*d^2*(1-2*p) + 2*b*d^2*(p-q) + 2*b*d*c*(p-1)_ + +2*b*p*q*(c+1) + (b^2 - 2*b)*p^2*d^2 + 2*b^2*p^2 + 4*b*(1-b)*p_ + + d^2*(p-1)^2 +--R +--R +--R (1) +--R 2 2 +--R (- 2c d + c )q +--R + +--R 2 2 2 +--R ((2b d + (- 2b + 2)c d + 2b c + 2b - 2)p - 2b d + 2c d - 2c - 2b + 2)q +--R + +--R 2 2 2 2 +--R ((b - 2b + 1)d + 2b + 2)p +--R + +--R 2 2 2 2 2 2 +--R ((- 2b + 2b - 2)d + (2b - 2)c d - 4b + 4b)p + (b + 1)d - 2b c d + c +--R + +--R 2 +--R 2b - 4b + 2 +--E 7 + +--S 8 +t2:=d*(2*p+1)*(q-p) + c*(p+2)*(1-q) + b*(b-2)*d + b*(1-2*b)*p*d_ + +b*c*(q+p-p*q-1) + b*(b+1)*p^2*d +--R +--R +--R (2) +--R 2 2 +--R ((2d + (- b - 1)c)p + d + (b - 2)c)q + (b + b - 2)d p +--R + +--R 2 2 +--R ((- 2b + b - 1)d + (b + 1)c)p + (b - 2b)d + (- b + 2)c +--E 8 + +--S 9 +t3:=-b^2*(p-1)^2 + 2*p*(p-q) - 2*(q-1) +--R +--R +--R 2 2 2 2 +--R (3) (- 2p - 2)q + (- b + 2)p + 2b p - b + 2 +--E 9 + +--S 10 +t4:=b^2 + 4*(p-q^2) + 3*c^2*(q-1)^2 - 3*d^2*(p-q)^2 + 3*b^2*d^2*(p-1)^2_ + +b^2*p*(p-2) + 6*b*d*c*(p+q+q*p-1) +--R +--R +--R (4) +--R 2 2 2 2 2 +--R (- 3d + 3c - 4)q + ((6d + 6b c d)p + 6b c d - 6c )q +--R + +--R 2 2 2 2 2 2 2 2 2 2 +--R ((3b - 3)d + b )p + (- 6b d + 6b c d - 2b + 4)p + 3b d - 6b c d + 3c +--R + +--R 2 +--R b +--E 10 + +@ +\section{CzaporGeddes3} +Paramters $a,b,c,d,e,f,g,h,k$ and variables $x,y$ +<<*>>= + +)clear all + +--S 11 +t1:=a*x^2+b*x*y+c*x+d*y^2+e*y+f +--R +--R +--R 2 2 +--R (1) d y + (b x + e)y + a x + c x + f +--E 11 + +--S 12 +t2:=b*x^2+4*d*x*y+2*e*x+g*y^2+h*y+k +--R +--R +--R 2 2 +--R (2) g y + (4d x + h)y + b x + 2e x + k +--E 12 + +@ +\section{CzaporGeddes4} +Paramters $a,b,c,d,e,f,g,h,k$ and variables $x,y,z$ +<<*>>= +)clear all + +--S 13 +t1:=x^2+a*y*z+d*x+g +--R +--R +--R 2 +--R (1) a y z + x + d x + g +--E 13 + +--S 14 +t2:=y^2+b*z*x+e*y+h +--R +--R +--R 2 +--R (2) b x z + y + e y + h +--E 14 + +--S 15 +t3:=z^2+c*x*y+f*z+k +--R +--R +--R 2 +--R (3) z + f z + c x y + k +--E 15 + +@ +\section{Hawes1} +Parameters $a,b$ and variables $x,A,B$ +<<*>>= +)clear all + +--S 16 +t1:=(x^2-A)^2 + (x^3+b*x-B)*(x^3+b*x-B-a)^2 +--R +--R +--R (1) +--R 9 7 6 2 5 4 +--R x + 3b x + (- 2a - 3B)x + 3b x + ((- 4a - 6B)b + 1)x +--R + +--R 3 2 2 3 2 2 2 2 +--R (b + a + 4B a + 3B )x + ((- 2a - 3B)b - 2A)x + (a + 4B a + 3B )b x +--R + +--R 2 2 3 2 +--R - B a - 2B a - B + A +--E 16 + +--S 17 +t2:=4*x*(x^2-A)+(3*x^2+b)*(x^3+b*x-B-a)*(3*(x^3+b*x-B)-a) +--R +--R +--R (2) +--R 8 6 5 2 4 3 +--R 9x + 21b x + (- 12a - 18B)x + 15b x + ((- 16a - 24B)b + 4)x +--R + +--R 3 2 2 2 2 2 2 +--R (3b + 3a + 12B a + 9B )x + ((- 4a - 6B)b - 4A)x + (a + 4B a + 3B )b +--E 17 + +--S 18 +t3:=12*x^2-4*A+6*x*(x^3+b*x-B-a)^2+4*(3*x^2+b)^2*(x^3+b*x-B-a)_ + +2*(x^3+b*x-B)*(3*x^2+b)^2+12*x*(x^3+b*x-B)*(x^3+b*x-B-a) +--R +--R +--R (3) +--R 7 5 4 2 3 2 +--R 72x + 126b x + (- 60a - 90B)x + 60b x + ((- 48a - 72B)b + 12)x +--R + +--R 3 2 2 2 +--R (6b + 6a + 24B a + 18B )x + (- 4a - 6B)b - 4A +--E 18 + +--S 19 +t4:=24*x+6*(x^3+b*x-B-a)^2+72*x*(x^3+b*x-B-a)*(3*x^2+b)+6*(3*x^2+b)^3_ + +36*x*(x^3+b*x-B)*(3*x^2+b)+12*(x^3+b*x-B)*(x^3+b*x-B-a) +--R +--R +--R (4) +--R 6 4 3 2 2 +--R 504x + 630b x + (- 240a - 360B)x + 180b x + ((- 96a - 144B)b + 24)x +--R + +--R 3 2 2 +--R 6b + 6a + 24B a + 18B +--E 19 + +@ +\section{Hawes2} +Parameters $a,b,c$ and variables $x,y[2],z[2]$ +<<*>>= +)clear all + +--S 20 +t1:=x+2*y1*z1+3*a*y1^2+5*y1^4+2*c*y1 +--R +--R +--R 4 2 +--R (1) 2y1 z1 + 5y1 + 3a y1 + 2c y1 + x +--E 20 + +--S 21 +t2:=x+2*y2*z2+3*a*y2^2+5*y2^4+2*c*y2 +--R +--R +--R 4 2 +--R (2) 2y2 z2 + 5y2 + 3a y2 + 2c y2 + x +--E 21 + +--S 22 +t3:=2*z2+6*a*y2+20*y2^3+2*c +--R +--R +--R 3 +--R (3) 2z2 + 20y2 + 6a y2 + 2c +--E 22 + +--S 23 +t4:=3*z1^2+y1^2+b +--R +--R +--R 2 2 +--R (4) 3z1 + y1 + b +--E 23 + +--S 24 +t5:=3*z2^2+y2^2+b +--R +--R +--R 2 2 +--R (5) 3z2 + y2 + b +--E 24 + +@ +\section{Hawes} +Paramters $a,b,c$ and variables $x,y[3],z[3]$ +<<*>>= +)clear all + +--S 25 +t1:=3*z1^2+y1^2+b +--R +--R +--R 2 2 +--R (1) 3z1 + y1 + b +--E 25 + +--S 26 +t2:=3*z1^2+y2^2+b +--R +--R +--R 2 2 +--R (2) 3z1 + y2 + b +--E 26 + +--S 27 +t3:=3*z3^2+y3^2+b +--R +--R +--R 2 2 +--R (3) 3z3 + y3 + b +--E 27 + +--S 28 +t4:=y1^2*z1+2*a*y1^3+4*y1^5+c*y1^2-z1^3-b*z1-y2^2*z2-2*a*y2^3_ + -4*y2^5-c*y2^2+z2^3+b*z2 +--R +--R +--R (4) +--R 3 2 3 2 5 3 2 5 +--R z2 + (- y2 + b)z2 - z1 + (y1 - b)z1 - 4y2 - 2a y2 - c y2 + 4y1 +--R + +--R 3 2 +--R 2a y1 + c y1 +--E 28 + +--S 29 +t5:=y2^2*z2+2*a*y2^3+4*y2^5+c*y2^2-z2^3-b*z2-y3^2*z3-2*a*y3^3_ + -4*y3^5-c*y3^2+z3^3+b*z3 +--R +--R +--R (5) +--R 3 2 3 2 5 3 2 5 +--R z3 + (- y3 + b)z3 - z2 + (y2 - b)z2 - 4y3 - 2a y3 - c y3 + 4y2 +--R + +--R 3 2 +--R 2a y2 + c y2 +--E 29 + +--S 30 +t6:=y3^2*z3+2*a*y3^3+4*y3^5+c*y3^2-z3^3-b*z3-y1^2*z1-2*a*y1^3_ + -4*y1^5-c*y1^2+z1^3+b*z1 +--R +--R +--R (6) +--R 3 2 3 2 5 3 2 5 +--R - z3 + (y3 - b)z3 + z1 + (- y1 + b)z1 + 4y3 + 2a y3 + c y3 - 4y1 +--R + +--R 3 2 +--R - 2a y1 - c y1 +--E 30 + +@ +\section{Hawes} +Paramters $a,b,c$ and variables $x,y[2],z[2]$ +<<*>>= +)clear all + +--S 31 +t1:=3*z1^2+y1^2+b +--R +--R +--R 2 2 +--R (1) 3z1 + y1 + b +--E 31 + +--S 32 +t2:=3*z2^2+y2^2+b +--R +--R +--R 2 2 +--R (2) 3z2 + y2 + b +--E 32 + +--S 33 +t3:=x+2*y1*z1+3*a*y1^2+5*y1^4+2*c*y1 +--R +--R +--R 4 2 +--R (3) 2y1 z1 + 5y1 + 3a y1 + 2c y1 + x +--E 33 + +--S 34 +t4:=x+2*y2*z2+3*a*y2^2+5*y2^4+2*c*y2 +--R +--R +--R 4 2 +--R (4) 2y2 z2 + 5y2 + 3a y2 + 2c y2 + x +--E 34 + +--S 35 +t5:=x*y1+z1^3+y1^2*z1+a*y1^3+y1^5+b*z1+c*y1^2-x*y2-z2^3-y2^2*z2_ + -a*y2^3-y2^5-b*z2-c*y2^2 +--R +--R +--R (5) +--R 3 2 3 2 5 3 2 +--R - z2 + (- y2 - b)z2 + z1 + (y1 + b)z1 - y2 - a y2 - c y2 - x y2 +--R + +--R 5 3 2 +--R y1 + a y1 + c y1 + x y1 +--E 35 + +--S 36 +t6:=(6*z1^2+18*a*z1*y1+6*y1-y1^3*z1+6*c*y1^2*z1-2*y1^2)_ + *(3*z2^2*y2+9*a*y2^2*z2+45*y2^4*z2-y2^3-3*x*z2+b*y2)_ + -(6*z2^2+18*a*z2*y2+60*y2^3*z2+6*c*y2^2*z2-2*y2^2)_ + *(3*z1^2*y1+9*a*y1^2*z1+45*y1^4*z1-y1^3-3*x*z1+b*y1) +--R +--R +--R (6) +--R 2 +--R (18y2 - 18y1)z1 +--R + +--R 3 2 4 2 +--R ((- 3y1 + 18c y1 + 54a y1)y2 - 270y1 - 54a y1 + 18x)z1 +--R + +--R 2 3 +--R (- 6y1 + 18y1)y2 + 6y1 - 6b y1 +--R * +--R 2 +--R z2 +--R + +--R 4 3 2 2 +--R (270y2 - 180y1 y2 + (- 18c y1 + 54a)y2 - 54a y1 y2 - 18x)z1 +--R + +--R 3 2 4 +--R (- 45y1 + 270c y1 + 810a y1)y2 +--R + +--R 4 2 3 +--R (- 2700y1 - 540a y1 + 180x)y2 +--R + +--R 4 3 2 2 +--R (- 270c y1 - 9a y1 + 162a y1 + 18c x)y2 +--R + +--R 4 2 2 3 2 +--R (- 810a y1 - 162a y1 + 54a x)y2 + 3x y1 - 18c x y1 - 54a x y1 +--R * +--R z1 +--R + +--R 2 4 3 3 +--R (- 90y1 + 270y1)y2 + (60y1 - 60b y1)y2 +--R + +--R 3 2 2 3 +--R (6c y1 - 18a y1 + (- 6b c + 54a)y1)y2 + (18a y1 - 18a b y1)y2 +--R + +--R 2 +--R 6x y1 - 18x y1 +--R * +--R z2 +--R + +--R 3 2 2 +--R (- 6y2 + 6y1 y2 + 6b y2)z1 +--R + +--R 3 2 3 4 2 2 +--R (y1 - 6c y1 - 18a y1)y2 + (90y1 + 18a y1 - 6x)y2 +--R + +--R 3 2 +--R (- b y1 + 6b c y1 + 18a b y1)y2 +--R * +--R z1 +--R + +--R 2 3 3 2 2 +--R (2y1 - 6y1)y2 + (- 2y1 + 2b y1)y2 + (- 2b y1 + 6b y1)y2 +--E 36 + +@ +\section{Reif} +Variables $x[16]$ +<<*>>= +)clear all + +--S 37 +t1:=x4*x13 + x5*x14 + x6*(1-x13-x14) +--R +--R +--R (1) (- x14 - x13 + 1)x6 + x14 x5 + x13 x4 +--E 37 + +--S 38 +t2:=x4*x15 + x5*x16 - x6*(x15+x16) +--R +--R +--R (2) (- x16 - x15)x6 + x16 x5 + x15 x4 +--E 38 + +--S 39 +t3:=x7*x13 + x8*x14 + x9*(1-x13-x14) +--R +--R +--R (3) (- x14 - x13 + 1)x9 + x14 x8 + x13 x7 +--E 39 + +--S 40 +t4:=x7*x15 + x8*x16 - x9*(x15+x16)-1 +--R +--R +--R (4) (- x16 - x15)x9 + x16 x8 + x15 x7 - 1 +--E 40 + +--S 41 +t5:=x10*x13 + x11*x14 + x12*(1-x13-x14) +--R +--R +--R (5) (- x12 + x11)x14 + (- x12 + x10)x13 + x12 +--E 41 + +--S 42 +t6:=x10*x15 + x11*x16 - x12*(x15+x16) +--R +--R +--R (6) (- x12 + x11)x16 + (- x12 + x10)x15 +--E 42 + +--S 43 +t7:=x1*x13 + x2*x14 + x3*(1-x13-x14) +--R +--R +--R (7) (- x14 - x13 + 1)x3 + x14 x2 + x1 x13 +--E 43 + +--S 44 +t8:=x1*x15 + x2*x16 - x3*(x15+x16) +--R +--R +--R (8) (- x16 - x15)x3 + x16 x2 + x1 x15 +--E 44 + +--S 45 +t9:=x1*x4*x13 + x2*x5*x14 + x3*x6*(1-x13-x14)-1 +--R +--R +--R (9) (- x14 - x13 + 1)x3 x6 + x14 x2 x5 + x1 x13 x4 - 1 +--E 45 + +--S 46 +t10:=x1*x4*x15 + x2*x5*x16 - x3*x6*(x15+x16) +--R +--R +--R (10) (- x16 - x15)x3 x6 + x16 x2 x5 + x1 x15 x4 +--E 46 + +--S 47 +t11:=x1*x7*x13 + x2*x8*x14 + x3*x9*(1-x13-x14) +--R +--R +--R (11) (- x14 - x13 + 1)x3 x9 + x14 x2 x8 + x1 x13 x7 +--E 47 + +--S 48 +t12:=x1*x7*x15 + x2*x8*x16 - x3*x9*(x15+x16) +--R +--R +--R (12) (- x16 - x15)x3 x9 + x16 x2 x8 + x1 x15 x7 +--E 48 + +--S 49 +t13:=x1*x10*x13 + x2*x11*x14 + x3*x12*(1-x13-x14) +--R +--R +--R (13) (- x12 x14 - x12 x13 + x12)x3 + x11 x14 x2 + x1 x10 x13 +--E 49 + +--S 50 +t14:=x1*x10*x15 + x2*x11*x16 - x3*x12*(x15+x16)-1 +--R +--R +--R (14) (- x12 x16 - x12 x15)x3 + x11 x16 x2 + x1 x10 x15 - 1 +--E 50 + +@ +\section{Reimer4} +Variables $x,y,z,t$ +<<*>>= +)clear all + +--S 51 +t1:=2*x^2-2*y^2+2*z^2-2*t^2-1 +--R +--R +--R 2 2 2 2 +--R (1) 2z - 2y + 2x - 2t - 1 +--E 51 + +--S 52 +t2:=2*x^3-2*y^3+2*z^3-2*t^3-1 +--R +--R +--R 3 3 3 3 +--R (2) 2z - 2y + 2x - 2t - 1 +--E 52 + +--S 53 +t3:=2*x^4-2*y^4+2*z^4-2*t^4-1 +--R +--R +--R 4 4 4 4 +--R (3) 2z - 2y + 2x - 2t - 1 +--E 53 + +--S 54 +t4:=2*x^5-2*y^5+2*z^5-2*t^5-1 +--R +--R +--R 5 5 5 5 +--R (4) 2z - 2y + 2x - 2t - 1 +--E 54 + +@ +\section{Reimer4} +Variables $x,y,z,t,u$ +<<*>>= +)clear all + +--S 55 +t1:=2*x^2-2*y^2+2*z^2-2*t^2+2*u^2-1 +--R +--R +--R 2 2 2 2 2 +--R (1) 2z - 2y + 2x + 2u - 2t - 1 +--E 55 + +--S 56 +t2:=2*x^3-2*y^3+2*z^3-2*t^3+2*u^3-1 +--R +--R +--R 3 3 3 3 3 +--R (2) 2z - 2y + 2x + 2u - 2t - 1 +--E 56 + +--S 57 +t3:=2*x^4-2*y^4+2*z^4-2*t^4+2*u^4-1 +--R +--R +--R 4 4 4 4 4 +--R (3) 2z - 2y + 2x + 2u - 2t - 1 +--E 57 + +--S 58 +t4:=2*x^5-2*y^5+2*z^5-2*t^5+2*u^5-1 +--R +--R +--R 5 5 5 5 5 +--R (4) 2z - 2y + 2x + 2u - 2t - 1 +--E 58 + +--S 59 +t5:=2*x^6-2*y^6+2*z^6-2*t^6+2*u^6-1 +--R +--R +--R 6 6 6 6 6 +--R (5) 2z - 2y + 2x + 2u - 2t - 1 +--E 59 + +@ +\section{Reimer(n)} +Given the variables $x[x]$ the general case of these +equations is given by: +\[-1/2+\sum_{i=1}^n{(-1)(i+1)x_i^k} \quad k=2..n+1\] + +\section{Gerdt} +Variables $x,y,z,t,u,v,w$ +<<*>>= +)clear all + +--S 60 +t1:=y*w-1/2*z*w+t*w +--R +--R +--R 1 +--R (1) - - w z + w y + t w +--R 2 +--E 60 + +--S 61 +t2:=-2/7*u*w^2+10/7*v*w^2-20/7*w^3+t*u-5*t*v+10*t*w +--R +--R +--R 20 3 10 2 2 +--R (2) - -- w + (-- v - - u)w + 10t w - 5t v + t u +--R 7 7 7 +--E 61 + +--S 62 +t3:=2/7*y*w^2-2/7*z*w^2+6/7*t*w^2-y*t+z*t-3*t^2 +--R +--R +--R 2 2 2 2 6 2 2 +--R (3) (- - w + t)z + (- w - t)y + - t w - 3t +--R 7 7 7 +--E 62 + +--S 63 +t4:=-2*v^3+4*u*v*w+5*v^2*w-6*u*w^2-7*v*w^2+15*w^3+42*y*v_ + -14*z*v-63*y*w+21*z*w-42*t*w+147*x +--R +--R +--R (4) +--R 3 2 +--R (21w - 14v)z + (- 63w + 42v)y + 147x + 15w + (- 7v - 6u)w +--R + +--R 2 3 +--R (5v + 4u v - 42t)w - 2v +--E 63 + +--S 64 +t5:=-9/7*u*w^3+45/7*v*w^3-135/7*w^4+2*z*v^2-2*t*v^2-4*z*u*w+10*t*u*w_ + -2*z*v*w-28*t*v*w+4*z*w^2+86*t*w^2-42*y*z+14*z^2+42*y*t_ + -14*z*t-21*x*u+105*x*v-315*x*w +--R +--R +--R (5) +--R 2 2 2 +--R 14z + (- 42y + 4w + (- 2v - 4u)w + 2v - 14t)z + 42t y +--R + +--R 135 4 45 9 3 2 +--R (- 315w + 105v - 21u)x - --- w + (-- v - - u)w + 86t w +--R 7 7 7 +--R + +--R 2 +--R (- 28t v + 10t u)w - 2t v +--E 64 + +--S 65 +t6:=6/7*y*w^3-9/7*z*w^3+36/7*t*w^3-2*x*v^2-4*y*t*w+6*z*t*w_ + -24*t^2*w+4*x*u*w+2*x*v*w-4*x*w^2+56*x*y-35*x*z+84*x*t +--R +--R +--R (6) +--R 9 3 6 3 +--R (- 35x - - w + 6t w)z + (56x + - w - 4t w)y +--R 7 7 +--R + +--R 2 2 36 3 2 +--R (- 4w + (2v + 4u)w - 2v + 84t)x + -- t w - 24t w +--R 7 +--E 65 + +--S 66 +t7:=2*u*v*w-6*v^2*w-u*w^2+13*v*w^2-5*w^3+14*y*w-28*t*w +--R +--R +--R 3 2 2 +--R (7) 14w y - 5w + (13v - u)w + (- 6v + 2u v - 28t)w +--E 66 + +--S 67 +t8:=u^2*w-3*u*v*w+5*u*w^2+14*y*w-28*t*w +--R +--R +--R 2 2 +--R (8) 14w y + 5u w + (- 3u v + u - 28t)w +--E 67 + +--S 68 +t9:=-2*z*u*w-2*t*u*w+4*y*v*w+6*z*v*w-2*t*v*w-16*y*w^2_ + -10*z*w^2+22*t*w^2+42*x*w +--R +--R +--R (9) +--R 2 2 2 +--R (- 10w + (6v - 2u)w)z + (- 16w + 4v w)y + 42w x + 22t w + (- 2t v - 2t u)w +--E 68 + +--S 69 +t10:=28/3*y*u*w+8/3*z*u*w-20/3*t*u*w-88/3*y*v*w-8*z*v*w_ + +68/3*t*v*w+52*y*w^2+40/3*z*w^2-44*t*w^2-84*x*w +--R +--R +--R (10) +--R 40 2 8 2 88 28 2 +--R (-- w + (- 8v + - u)w)z + (52w + (- -- v + -- u)w)y - 84w x - 44t w +--R 3 3 3 3 +--R + +--R 68 20 +--R (-- t v - -- t u)w +--R 3 3 +--E 69 + +--S 70 +t11:=-4*y*z*w+10*y*t*w+8*z*t*w-20*t^2*w+12*x*u*w-30*x*v*w+15*x*w^2 +--R +--R +--R 2 2 +--R (11) (- 4w y + 8t w)z + 10t w y + (15w + (- 30v + 12u)w)x - 20t w +--E 70 + +--S 71 +t12:=-y^2*w+1/2*y*z*w+y*t*w-z*t*w+2*t^2*w-3*x*u*w+6*x*v*w-3*x*w^2 +--R +--R +--R 1 2 2 2 +--R (12) (- w y - t w)z - w y + t w y + (- 3w + (6v - 3u)w)x + 2t w +--R 2 +--E 71 + +--S 72 +t13:=8*x*y*w-4*x*z*w+8*x*t*w +--R +--R +--R (13) - 4w x z + 8w x y + 8t w x +--E 72 + +@ +\section{Gerdt2} +Variables $x,y,z,t,u$ +<<*>>= +)clear all + +--S 73 +t1:=35*y^4-30*x*y^2-210*y^2*z+3*x^2+30*x*z-105*z^2+140*y*t-21*u +--R +--R +--R 2 2 4 2 2 +--R (1) - 105z + (- 210y + 30x)z + 35y - 30x y + 140t y + 3x - 21u +--E 73 + +--S 74 +t2:=5*x*y^3-140*y^3*z-3*x^2*y+45*x*y*z-420*y*z^2+210*y^2*t_ + -25*x*t+70*z*t+126*y*u +--R +--R +--R (2) +--R 2 3 3 2 2 +--R - 420y z + (- 140y + 45x y + 70t)z + 5x y + 210t y + (- 3x + 126u)y +--R + +--R - 25t x +--E 74 + +@ +\section{Gerdt3} +Variables $x,y,z,t$ +<<*>>= +)clear all + +--S 75 +t1:=6*x*y^2*t-x^2*z*t-6*x*y*z*t+3*x*z^2*t-2*z^3*t-6*x*y^2+6*x*y*z-2*x*z^2 +--R +--R +--R 3 2 2 2 +--R (1) - 2t z + (3t - 2)x z + ((- 6t + 6)x y - t x )z + (6t - 6)x y +--E 75 + +--S 76 +t2:=-63*x*y^2*t^2+9*x^2*z*t^2+63*x*y*z*t^2+18*y^2*z*t^2-27*x*z^2*t^2_ + -18*y*z^2*t^2+18*z^3*t^2+78*x*y^2*t-78*x*y*z*t-18*y^2*z*t_ + +24*x*z^2*t+18*y*z^2*t-9*z^3*t-15*x*y^2+15*x*y*z-5*x*z^2 +--R +--R +--R (2) +--R 2 3 2 2 2 +--R (18t - 9t)z + ((- 18t + 18t)y + (- 27t + 24t - 5)x)z +--R + +--R 2 2 2 2 2 2 2 +--R ((18t - 18t)y + (63t - 78t + 15)x y + 9t x )z + (- 63t + 78t - 15)x y +--E 76 + +--S 77 +t3:=18*x^2*y^2*t-3*x^3*z*t-18*x^2*y*z*t+12*x*y^2*z*t+5*x^2*z^2*t_ + -12*x*y*z^2*t+6*x*z^3*t-8*z^4*t-18*x^2*y^2+18*x^2*y*z-12*x*y^2*z_ + -4*x^2*z^2+12*x*y*z^2-6*x*z^3 +--R +--R +--R (3) +--R 4 3 2 2 +--R - 8t z + (6t - 6)x z + ((- 12t + 12)x y + (5t - 4)x )z +--R + +--R 2 2 3 2 2 +--R ((12t - 12)x y + (- 18t + 18)x y - 3t x )z + (18t - 18)x y +--E 77 + +--S 78 +t4:=-x^2*y*t+3*x*y^2*t+10*y^3*t-15*y^2*z*t+3*y*z^2*t-3*x*y^2-10*y^3+x*y*z_ + +15*y^2*z-5*y*z^2 +--R +--R +--R (4) +--R 2 2 3 2 2 +--R (3t - 5)y z + ((- 15t + 15)y + x y)z + (10t - 10)y + (3t - 3)x y - t x y +--E 78 + +@ +\section{Gerdt4} +Variables $x,y,z,t,u,v,w,a,b,c$ +<<*>>= +)clear all + +--S 79 +t1:=y*t-y*u-u*b+u*c +--R +--R +--R (1) (- u + t)y + (c - b)u +--E 79 + +--S 80 +t2:=2*x*y^2*t-x*y^2*u-2*y^2*t*v+y^2*u*v-x*y*z*a+12*x*t^2*a-4*x*t*u*a_ + -x*u^2*a+y*z*v*a-2*t*u*v*a+u^2*v*a-x*u*w*a+u*v*w*a-6*x*z*a*b +--R +--R +--R (2) +--R 2 +--R ((- a x + a v)y - 6a b x)z + ((- u + 2t)x + (u - 2t)v)y +--R + +--R 2 2 2 +--R (- a u w - a u - 4a t u + 12a t )x + a u v w + (a u - 2a t u)v +--E 80 + +--S 81 +t3:=x*y^2*z-y^2*z*v+6*x*z*t*a+x*z*u*a-z*u*v*a-2*x*y*z*b+2*y*z*v*b_ + -2*x*u*w*b+2*u*v*w*b-12*x*z*b^2+x*y*z*c-y*z*v*c+x*u*w*c_ + -u*v*w*c+6*x*z*b*c +--R +--R +--R (3) +--R 2 2 +--R (x - v)y + ((c - 2b)x + (- c + 2b)v)y + (a u + 6a t + 6b c - 12b )x +--R + +--R - a u v +--R * +--R z +--R + +--R (c - 2b)u w x + (- c + 2b)u v w +--E 81 + +--S 82 +t4:=x*y*u-y*u*v+3*x*z*a+3*x*t*b+x*u*b-u*v*b +--R +--R +--R (4) 3a x z + (u x - u v)y + (b u + 3b t)x - b u v +--E 82 + +--S 83 +t5:=5*x^2*y*t-5*x^2*y*u-10*x*y*t*v+10*x*y*u*v+5*y*t*v^2-5*y*u*v^2_ + -6*x^2*z*a-12*x*z*v*a+4*x^2*t*b-7*x^2*u*b+16*x*t*v*b+8*x*u*v*b_ + -2*t*v^2*b-u*v^2*b+8*x^2*t*c+x^2*u*c-10*x*t*v*c-2*x*u*v*c_ + +2*t*v^2*c+u*v^2*c +--R +--R +--R (5) +--R 2 2 2 +--R (- 6a x - 12a v x)z + ((- 5u + 5t)x + (10u - 10t)v x + (- 5u + 5t)v )y +--R + +--R 2 +--R ((c - 7b)u + (8c + 4b)t)x + ((- 2c + 8b)u + (- 10c + 16b)t)v x +--R + +--R 2 +--R ((c - b)u + (2c - 2b)t)v +--E 83 + +--S 84 +t6:=-9*x^4*t*v*c+9*x^4*u*v*c-18*x^3*t*v^2*c-9*x^3*u*v^2*c+3*x^4*y*t_ + -4*x^4*y*u-9*x^3*y*t*v+10*x^3*y*u*v+9*x^2*y*t*v^2-6*x^2*y*u*v^2_ + -3*x*y*t*v^3-2*x*y*u*v^3+2*y*u*v^4-6*x^4*z*a-45*x^3*z*v*a_ + -27*x^2*z*v^2*a-3*x*z*v^3*a-6*x^4*t*b-2*x^4*u*b-45*x^3*t*v*b_ + +32*x^3*u*v*b-27*x^2*t*v^2*b-30*x^2*u*v^2*b-3*x*t*v^3*b-x*u*v^3*b+u*v^4*b +--R +--R +--R (6) +--R 4 3 2 2 3 +--R (- 6a x - 45a v x - 27a v x - 3a v x)z +--R + +--R 4 3 2 2 3 +--R (- 4u + 3t)x + (10u - 9t)v x + (- 6u + 9t)v x + (- 2u - 3t)v x +--R + +--R 4 +--R 2u v +--R * +--R y +--R + +--R 4 +--R ((9c u - 9c t)v - 2b u - 6b t)x +--R + +--R 2 3 2 2 +--R ((- 9c u - 18c t)v + (32b u - 45b t)v)x + (- 30b u - 27b t)v x +--R + +--R 3 4 +--R (- b u - 3b t)v x + b u v +--E 84 + +--S 85 +t7:=w*b-t*c+u*c-w*c +--R +--R +--R (7) (- c + b)w + c u - c t +--E 85 + +--S 86 +t8:=-6*z*t*v*a+x*z*w*a-z*v*w*a-2*x*w^2*b+2*v*w^2*b+12*z*v*b^2_ + +x*y*z*c-y*z*v*c+x*w^2*c-v*w^2*c-2*x*z*b*c-4*z*v*b*c+x*z*c^2-z*v*c^2 +--R +--R +--R (8) +--R 2 2 2 +--R ((c x - c v)y + (a w + c - 2b c)x - a v w + (- 6a t - c - 4b c + 12b )v)z +--R + +--R 2 2 +--R (c - 2b)w x + (- c + 2b)v w +--E 86 + +--S 87 +t9:=-12*t^2*v*a+6*t*u*v*a+2*x*t*w*a-x*u*w*a-2*t*v*w*a+u*v*w*a_ + -x*w^2*a+v*w^2*a+6*z*v*a*b+2*x*y*t*c-x*y*u*c-2*y*t*v*c+y*u*v*c_ + -x*z*a*c+z*v*a*c +--R +--R +--R (9) +--R (- a c x + (a c + 6a b)v)z + ((- c u + 2c t)x + (c u - 2c t)v)y +--R + +--R 2 2 2 +--R (- a w + (- a u + 2a t)w)x + a v w + (a u - 2a t)v w + (6a t u - 12a t )v +--E 87 + +--S 88 +t10:=3*z*v*a+3*t*v*b-x*t*c+t*v*c-x*w*c+v*w*c +--R +--R +--R (10) 3a v z + (- c w - c t)x + c v w + (c + 3b)t v +--E 88 + +--S 89 +t11:=-12*x*z*v*a-6*z*v^2*a-2*x^2*t*b+2*x^2*u*b+16*x*t*v*b-10*x*u*v*b_ + +4*t*v^2*b+8*u*v^2*b+5*x^2*w*b-10*x*v*w*b+5*v^2*w*b-x^2*t*c_ + +x^2*u*c+8*x*t*v*c-2*x*u*v*c-7*t*v^2*c+u*v^2*c-5*x^2*w*c_ + +10*x*v*w*c-5*v^2*w*c +--R +--R +--R (11) +--R 2 2 +--R (- 12a v x - 6a v )z + ((- 5c + 5b)w + (c + 2b)u + (- c - 2b)t)x +--R + +--R 2 +--R ((10c - 10b)v w + ((- 2c - 10b)u + (8c + 16b)t)v)x + (- 5c + 5b)v w +--R + +--R 2 +--R ((c + 8b)u + (- 7c + 4b)t)v +--E 89 + +--S 90 +t12:=-18*x^2*u*v^3*b-9*x*u*v^4*b-9*x^2*u*v^3*c+9*x*u*v^4*c-3*x^3*z*v*a_ + -27*x^2*z*v^2*a-45*x*z*v^3*a-6*z*v^4*a-3*x^3*t*v*b_ + -27*x^2*t*v^2*b-45*x*t*v^3*b-6*t*v^4*b-3*x^3*v*w*b_ + +9*x^2*v^2*w*b-9*x*v^3*w*b+3*v^4*w*b+x^4*t*c-x^3*t*v*c_ + -30*x^2*t*v^2*c+32*x*t*v^3*c-2*t*v^4*c+2*x^4*w*c-2*x^3*v*w*c_ + -6*x^2*v^2*w*c+10*x*v^3*w*c-4*v^4*w*c +--R +--R +--R (12) +--R 3 2 2 3 4 4 +--R (- 3a v x - 27a v x - 45a v x - 6a v )z + (2c w + c t)x +--R + +--R 3 +--R ((- 2c - 3b)v w + (- c - 3b)t v)x +--R + +--R 2 3 2 2 +--R ((- 6c + 9b)v w + (- 9c - 18b)u v + (- 30c - 27b)t v )x +--R + +--R 3 4 3 4 +--R ((10c - 9b)v w + (9c - 9b)u v + (32c - 45b)t v )x + (- 4c + 3b)v w +--R + +--R 4 +--R (- 2c - 6b)t v +--E 90 + +@ +\section{Gerdt5} +Varables $x,y,z,t,u,v,a,A,B,C,D,E,F$ +<<*>>= +)clear all + +--S 91 +t1:=v*A +--R +--R +--R (1) A v +--E 91 + +--S 92 +t2:=u*A+14*B +--R +--R +--R (2) A u + 14B +--E 92 + +--S 93 +t3:=z*A +--R +--R +--R (3) A z +--E 93 + +--S 94 +t4:=u*a*A+3*z*A+2*t*A+168*B +--R +--R +--R (4) 3A z + A a u + 2A t + 168B +--E 94 + +--S 95 +t5:=y*A+5*u*B +--R +--R +--R (5) A y + 5B u +--E 95 + +--S 96 +t6:=5*v*C+21*D +--R +--R +--R (6) 5C v + 21D +--E 96 + +--S 97 +t7:=10*u*C+14*E +--R +--R +--R (7) 10C u + 14E +--E 97 + +--S 98 +t8:=-5*y*C-u*E+105*F +--R +--R +--R (8) - 5C y - E u + 105F +--E 98 + +--S 99 +t9:=5*z*C+2*u*D +--R +--R +--R (9) 5C z + 2D u +--E 99 + +--S 100 +t10:=-2/7*v^2+t-4*u-A +--R +--R +--R 2 2 +--R (10) - - v - 4u + t - A +--R 7 +--E 100 + +--S 101 +t11:=-2/7*u^2+y-B +--R +--R +--R 2 2 +--R (11) y - - u - B +--R 7 +--E 101 + +--S 102 +t12:=7*u-C +--R +--R +--R (12) 7u - C +--E 102 + +--S 103 +t13:=3/7*v^3-2*t*v+6*u*v-7*z-D +--R +--R +--R 3 3 +--R (13) - 7z + - v + (6u - 2t)v - D +--R 7 +--E 103 + +--S 104 +t14:=9/7*u*v^2-2*t*u+16*u^2-6*z*v-42*y-E +--R +--R +--R 9 2 2 +--R (14) - 6v z - 42y + - u v + 16u - 2t u - E +--R 7 +--E 104 + +--S 105 +t15:=3/7*u^3-2*y*u+7*x-F +--R +--R +--R 3 3 +--R (15) - 2u y + 7x + - u - F +--R 7 +--E 105 + +@ +\section{Gertd6} +Variables $x,y,z,t$ +<<*>>= +)clear all + +--S 106 +t1:=-2*y^3*z+6*x^2*z*t-6*x*y*z*t+3*y^2*z*t-y*z*t^2-6*x^2*t+6*x*y*t-2*y^2*t +--R +--R +--R 3 2 2 2 2 2 +--R (1) (- 2y + 3t y + (- 6t x - t )y + 6t x )z - 2t y + 6t x y - 6t x +--E 106 + +--S 107 +t1:=18*x^2*y*z^2-18*x*y^2*z^2+18*y^2*z^2-63*x^2*z^2*t+63*x*y*z^2*t_ + -27*y^2*z^2*t+9*y*z^2*t^2-18*x^2*y*z+18*x*y^2*z-9*y^3*z_ + +78*x^2*z*t-78*x*y*z*t+24*y^2*z*t-15*x^2*t+15*x*y*t-5*y^2*t +--R +--R +--R (2) +--R 2 2 2 2 2 +--R ((- 18x - 27t + 18)y + (18x + 63t x + 9t )y - 63t x )z +--R + +--R 3 2 2 2 2 +--R (- 9y + (18x + 24t)y + (- 18x - 78t x)y + 78t x )z - 5t y + 15t x y +--R + +--R 2 +--R - 15t x +--E 107 + +--S 108 +t1:=-8*y^4*z+12*x^2*y*z*t-12*x*y^2*z*t+6*y^3*z*t+18*x^2*z*t^2_ + -18*x*y*z*t^2+5*y^2*z*t^2-3*y*z*t^3-12*x^2*y*t+12*x*y^2*t_ + -6*y^3*t-18*x^2*t^2+18*x*y*t^2-4*y^2*t^2 +--R +--R +--R (3) +--R 4 3 2 2 2 2 3 2 2 +--R (- 8y + 6t y + (- 12t x + 5t )y + (12t x - 18t x - 3t )y + 18t x )z +--R + +--R 3 2 2 2 2 2 2 +--R - 6t y + (12t x - 4t )y + (- 12t x + 18t x)y - 18t x +--E 108 + +--S 109 +t1:=10*x^3*z-15*x^2*y*z+3*x*y^3*z+3*x^2*z*t-x*z*t^2-10*x^3+15*x^2*y_ + -5*x*y^2-3*x^2*t+x*y*t +--R +--R +--R (4) +--R 3 2 3 2 2 2 2 3 2 +--R (3x y - 15x y + 10x + 3t x - t x)z - 5x y + (15x + t x)y - 10x - 3t x +--E 109 + +@ +\section{Hairer1} +Variables $a,b,c,d,e,f,g,h$ +<<*>>= +)clear all + +--S 110 +t1:=a-f +--R +--R +--R (1) - f + a +--E 110 + +--S 111 +t2:=b-g-h +--R +--R +--R (2) - h - g + b +--E 111 + +--S 112 +t3:=c+d+e-1 +--R +--R +--R (3) e + d + c - 1 +--E 112 + +--S 113 +t4:=b*c+a*d-1/2 +--R +--R +--R 1 +--R (4) a d + b c - - +--R 2 +--E 113 + +--S 114 +t5:=b^2*c+a^2*d-1/3 +--R +--R +--R 2 2 1 +--R (5) a d + b c - - +--R 3 +--E 114 + +--S 115 +t6:=a*c*g-1/6 +--R +--R +--R 1 +--R (6) a c g - - +--R 6 +--E 115 + +@ +\section{Hairer2} +Variables $a,b,c,d,e,f,g,h,i,j,k,l,m$ +<<*>>= +)clear all + +--S 116 +t1:=d+e+f+g-1 +--R +--R +--R (1) g + f + e + d - 1 +--E 116 + +--S 117 +t2:=c*d+b*e+a*f-1/2 +--R +--R +--R 1 +--R (2) a f + b e + c d - - +--R 2 +--E 117 + +--S 118 +t3:=c^2*d+b^2*e+a^2*f-1/3 +--R +--R +--R 2 2 2 1 +--R (3) a f + b e + c d - - +--R 3 +--E 118 + +--S 119 +t4:=a*e*i+a*d*l+b*d*m-1/6 +--R +--R +--R 1 +--R (4) b d m + a d l + a e i - - +--R 6 +--E 119 + +--S 120 +t5:=c^3*d+b^3*e+a^3*f-1/4 +--R +--R +--R 3 3 3 1 +--R (5) a f + b e + c d - - +--R 4 +--E 120 + +--S 121 +t6:=a*b*e*i+a*c*d*l+b*c*d*m-1/8 +--R +--R +--R 1 +--R (6) b c d m + a c d l + a b e i - - +--R 8 +--E 121 + +--S 122 +t7:=a^2*e*i+a^2*d*l+b^2*d*m-1/2 +--R +--R +--R 2 2 2 1 +--R (7) b d m + a d l + a e i - - +--R 2 +--E 122 + +--S 123 +t8:=a*d*i*m-1/24 +--R +--R +--R 1 +--R (8) a d i m - -- +--R 24 +--E 123 + +--S 124 +t9:=a-h +--R +--R +--R (9) - h + a +--E 124 + +--S 125 +t10:=b-i-j +--R +--R +--R (10) - j - i + b +--E 125 + +--S 126 +t11:=c-k-l-m +--R +--R +--R (11) - m - l - k + c +--E 126 + +@ +\section{Hairer3} +Variables $a,b,c,d,e,f,g,h,i,j,k,l,m,n$ +<<*>>= +)clear all + +--S 127 +t1:=a*e+b*f+c*g+d*h-1/2 +--R +--R +--R 1 +--R (1) d h + c g + b f + a e - - +--R 2 +--E 127 + +--S 128 +t2:=a^2*e+b^2*f+c^2*g+d^2*h-1/3 +--R +--R +--R 2 2 2 2 1 +--R (2) d h + c g + b f + a e - - +--R 3 +--E 128 + +--S 129 +t3:=a*f*i+a*g*j+b*g*k+a*h*l+b*h*m+c*h*n-1/6 +--R +--R +--R 1 +--R (3) c h n + b h m + a h l + b g k + a g j + a f i - - +--R 6 +--E 129 + +--S 130 +t4:=a^3*e+b^3*f+c^3*g+d^3*h-1/4 +--R +--R +--R 3 3 3 3 1 +--R (4) d h + c g + b f + a e - - +--R 4 +--E 130 + +--S 131 +t5:=a*b*f*i+a*c*g*j+b*c*g*k+a*d*h*l+b*d*h*m+c*d*h*n-1/8 +--R +--R +--R 1 +--R (5) c d h n + b d h m + a d h l + b c g k + a c g j + a b f i - - +--R 8 +--E 131 + +--S 132 +t6:=a^2*f*i+a^2*g*j+b^2*g*k+a^2*h*l+b^2*h*m+c^2*h*n-1/12 +--R +--R +--R 2 2 2 2 2 2 1 +--R (6) c h n + b h m + a h l + b g k + a g j + a f i - -- +--R 12 +--E 132 + +--S 133 +t7:=a*g*i*k+a*h*i*m+a*h*j*n+b*h*k*n-1/24 +--R +--R +--R 1 +--R (7) (b h k + a h j)n + a h i m + a g i k - -- +--R 24 +--E 133 + +--S 134 +t8:=a^4*e+b^4*f+c^4*g+d^4*h-1/5 +--R +--R +--R 4 4 4 4 1 +--R (8) d h + c g + b f + a e - - +--R 5 +--E 134 + +--S 135 +t9:=a*b^2*f*i+a*c^2*g*j+b*c^2*g*k+ad^2*h*l+b*d^2*h*m+c*d^2*h*n-1/10 +--R +--R +--R 2 2 2 2 2 2 1 +--R (9) c d h n + b d h m + ad h l + b c g k + a c g j + a b f i - -- +--R 10 +--E 135 + +--S 136 +t10:=a^2*b*f*i+a^2*c*g*j+b^2*c*g*k+a^3*h*l+b^2*d*h*m+c^2*d*h*n-1/15 +--R +--R +--R 2 2 3 2 2 2 1 +--R (10) c d h n + b d h m + a h l + b c g k + a c g j + a b f i - -- +--R 15 +--E 136 + +--S 137 +t11:=a*c*g*i*k+a*d*h*i*m+a*d*h*j*n+b*d*h*k*n-1/30 +--R +--R +--R 1 +--R (11) (b d h k + a d h j)n + a d h i m + a c g i k - -- +--R 30 +--E 137 + +--S 138 +t12:=a^2*f*i^2+a^2*g*j^2+2*a*b*g*j*k+b^2*g*k^2+a^2*h*l^2+2*a*b*h*l*m + +b^2*h*m^2+2*a*c*h*l*n+2*b*c*h*m*n+c^2*h*n^2-1/20 +--R +--R +--R (12) +--R 2 2 2 2 2 2 2 2 +--R c h n + (2b c h m + 2a c h l)n + b h m + 2a b h l m + a h l + b g k +--R + +--R 2 2 2 2 1 +--R 2a b g j k + a g j + a f i - -- +--R 20 +--E 138 + +--S 139 +t13:=a^2*f*i+a^3*g*j+b^3*g*k+a^3*h*l+b^3*h*m+c^3*h*n-1/20 +--R +--R +--R 3 3 3 3 3 2 1 +--R (13) c h n + b h m + a h l + b g k + a g j + a f i - -- +--R 20 +--E 139 + +--S 140 +t14:=a*b*g*i*k+a*b*h*i*m+a*c*h*j*n+b*c*h*k*n-1/40 +--R +--R +--R 1 +--R (14) (b c h k + a c h j)n + a b h i m + a b g i k - -- +--R 40 +--E 140 + +--S 141 +t15:=a^2*g*i*k+a^2*h*i*m+a^2*h*j*n+b^2*h*k*n-1/60 +--R +--R +--R 2 2 2 2 1 +--R (15) (b h k + a h j)n + a h i m + a g i k - -- +--R 60 +--E 141 + +--S 142 +t16:=a*h*i*k*n-1/120 +--R +--R +--R 1 +--R (16) a h i k n - --- +--R 120 +--E 142 + +@ +\section{Hairer4} +Variables $a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t$ +<<*>>= +)clear all + +--S 143 +t1:=a*f+b*g+c*h+d*i+e*j-1/2 +--R +--R +--R 1 +--R (1) e j + d i + c h + b g + a f - - +--R 2 +--E 143 + +--S 144 +t2:=a^2*f+b^2*g+c^2*h+d^2*i+e^2*j-1/3 +--R +--R +--R 2 2 2 2 2 1 +--R (2) e j + d i + c h + b g + a f - - +--R 3 +--E 144 + +--S 145 +t3:=t*d*j+a*g*k+a*h*l+b*h*m+a*i*n+b*i*o+c*i*p+a*j*q+b*j*r+c*j*s-1/6 +--R +--R +--R (3) +--R d j t + c j s + b j r + a j q + c i p + b i o + a i n + b h m + a h l +--R + +--R 1 +--R a g k - - +--R 6 +--E 145 + +--S 146 +t4:=a^3*f+b^3*g+c^3*h+d^3*i+e^3*j-1/4 +--R +--R +--R 3 3 3 3 3 1 +--R (4) e j + d i + c h + b g + a f - - +--R 4 +--E 146 + +--S 147 +t5:=t*d*e*j+a*b*g*k+a*c*h*l+b*c*h*m+a*d*i*n+b*d*i*o+c*d*i*p+a*e*j*q_ + +b*e*j*r+c*e*j*s-1/8 +--R +--R +--R (5) +--R d e j t + c e j s + b e j r + a e j q + c d i p + b d i o + a d i n +--R + +--R 1 +--R b c h m + a c h l + a b g k - - +--R 8 +--E 147 + +--S 148 +t6:=t*d^2*j+a^2*g*k+a^2*h*l+b^2*h*m+a^2*i*n+b^2*i*o+c^2*i*p+a^2*j*g_ + +b^2*j*r+c^2*j*s-1/12 +--R +--R +--R (6) +--R 2 2 2 2 2 2 2 2 2 +--R d j t + c j s + b j r + c i p + b i o + a i n + b h m + a h l + a g k +--R + +--R 2 1 +--R a g j - -- +--R 12 +--E 148 + +--S 149 +t7:=a*h*k*m+t*a*j*n+t*b*j*o+a*i*k*o+t*c*j*p+a*i*l*p+b*i*m*p+a*j*k*r_ + +a*j*l*s+b*j*m*s-1/24 +--R +--R +--R (7) +--R (c j p + b j o + a j n)t + (b j m + a j l)s + a j k r + (b i m + a i l)p +--R + +--R 1 +--R a i k o + a h k m - -- +--R 24 +--E 149 + +--S 150 +t8:=a^4*f+b^4*g+c^4*h+d^4*i+e^4*j-1/5 +--R +--R +--R 4 4 4 4 4 1 +--R (8) e j + d i + c h + b g + a f - - +--R 5 +--E 150 + +--S 151 +t9:=t*d*e^2*j+a*b^2*g*k+a*c^2*h*l+b*c^2*h*m+a*d^2*i*n+b*d^2*i*o_ + +c*d^2*i*p+a*e^2*j*g+b*e^2*j*r+c*e^2*j*s-1/10 +--R +--R +--R (9) +--R 2 2 2 2 2 2 2 +--R d e j t + c e j s + b e j r + c d i p + b d i o + a d i n + b c h m +--R + +--R 2 2 2 1 +--R a c h l + a b g k + a e g j - -- +--R 10 +--E 151 + +--S 152 +t10:=t*d^2*e*j+a^2*b*g*k+a^2*c*h*l+b^2*c*h*m+a^2*d*i*n+b^2*d*i*o_ + +c^2*d*i*p+a^2*e*j*q+b^2*e*j*r+c^2*e*j*s-1/15 +--R +--R +--R (10) +--R 2 2 2 2 2 2 2 +--R d e j t + c e j s + b e j r + a e j q + c d i p + b d i o + a d i n +--R + +--R 2 2 2 1 +--R b c h m + a c h l + a b g k - -- +--R 15 +--E 152 + +--S 153 +t11:=a*c*h*k*m+t*a*e*j*n+t*b*e*j*o+a*d*i*k*o+t*c*e*j*p+a*d*i*l*p_ + +b*d*i*m*p+a*e*j*k*r+a*e*j*l*s+b*e*j*m*s-1/30 +--R +--R +--R (11) +--R (c e j p + b e j o + a e j n)t + (b e j m + a e j l)s + a e j k r +--R + +--R 1 +--R (b d i m + a d i l)p + a d i k o + a c h k m - -- +--R 30 +--E 153 + +--S 154 +t12:=t^2*d^2*j+a^2*g*k^2+a^2*h*l^2+2*a*b*h*l*m+b^2*h*m^2+a^2*i*n^2_ + +2*a*b*i*n*o+b^2*i*o^2+2*a*c*i*n*p+2*b*c*i*o*p+c^2*i*p^2_ + +2*t*a*d*j*q+a^2*j*q^2+2*t*b*d*j*r+2*a*b*j*q*r+b^2*j*r^2_ + +2*t*c*d*j*s+2*a*c*j*q*s+2*b*c*j*r*s+c^2*j*s^2-1/20 +--R +--R +--R (12) +--R 2 2 2 2 +--R d j t + (2c d j s + 2b d j r + 2a d j q)t + c j s +--R + +--R 2 2 2 2 2 2 +--R (2b c j r + 2a c j q)s + b j r + 2a b j q r + a j q + c i p +--R + +--R 2 2 2 2 2 2 +--R (2b c i o + 2a c i n)p + b i o + 2a b i n o + a i n + b h m + 2a b h l m +--R + +--R 2 2 2 2 1 +--R a h l + a g k - -- +--R 20 +--E 154 + +--S 155 +t13:=t*d^3*j+a^3*g*k+a^3*h*l+b^3*h*m+a^3*i*n+b^3*i*o+c^3*i*p_ + +a^3*j*q+b^3*j*r+c^3*j*s-1/20 +--R +--R +--R (13) +--R 3 3 3 3 3 3 3 3 3 +--R d j t + c j s + b j r + a j q + c i p + b i o + a i n + b h m + a h l +--R + +--R 3 1 +--R a g k - -- +--R 20 +--E 155 + +--S 156 +t14:=a*b*h*k*m+t*a*d*j*n+t*b*d*j*o+a*b*i*k*o+t*c*d*j*p+a*c*i*l*p_ + +b*c*i*m*p+a*b*j*k*r+a*c*j*l*s+b*c*j*m*s-1/40 +--R +--R +--R (14) +--R (c d j p + b d j o + a d j n)t + (b c j m + a c j l)s + a b j k r +--R + +--R 1 +--R (b c i m + a c i l)p + a b i k o + a b h k m - -- +--R 40 +--E 156 + +--S 157 +t15:=a^2*h*k*m+t*a^2*j*n+t*b^2*j*o+a^2*i*k*o+t*c^2*j*p+a^2*i*l*p_ + +b^2*i*m*p+a^2*j*k*r+a^2*j*l*s+b^2*j*m*s-1/60 +--R +--R +--R (15) +--R 2 2 2 2 2 2 2 2 +--R (c j p + b j o + a j n)t + (b j m + a j l)s + a j k r + (b i m + a i l)p +--R + +--R 2 2 1 +--R a i k o + a h k m - -- +--R 60 +--E 157 + +--S 158 +t16:=t*a*j*k*o+t*a*j*l*p+t*b*j*m*p+a*i*k*m*p+a*j*k*m*s-1/20 +--R +--R +--R 1 +--R (16) ((b j m + a j l)p + a j k o)t + a j k m s + a i k m p - -- +--R 20 +--E 158 + +@ +\section{Hietarinta1} +Variables $a,c,j,l,m,n,p,v,g,h$ +<<*>>= +)clear all + +--S 159 +t1:=c^2*p-a*c+c*l+a-p-h +--R +--R +--R 2 +--R (1) (c - 1)p + c l - h - a c + a +--E 159 + +--S 160 +t2:=a*c*h+c^2+c*n+m*h +--R +--R +--R 2 +--R (2) c n + h m + a c h + c +--E 160 + +--S 161 +t3:=-a^2*c+a*c*l+a*c*g-c*l*h+a^2+2*c^2-a*m-a*h+l*h-2 +--R +--R +--R 2 2 2 +--R (3) - a m + ((- c + 1)h + a c)l - a h + a c g + 2c - a c + a - 2 +--E 161 + +--S 162 +t4:=-a*c^2+a*c*j-c^2*m+a*c*n+c^2*v-c*n*h-c*m+n*h +--R +--R +--R 2 2 2 +--R (4) c v + ((- c + 1)h + a c)n + (- c - c)m + a c j - a c +--E 162 + +--S 163 +t5:=-c*l*g-a*l-j-1 +--R +--R +--R (5) (- c g - a)l - j - 1 +--E 163 + +--S 164 +t6:=-c*n*g-c*l-c*m+j*m+c*g+c*h +--R +--R +--R (6) - c g n + (j - c)m - c l + c h + c g +--E 164 + +--S 165 +t7:=-c*j*l-a*j+j*l-a*n+c*g+a-v+g +--R +--R +--R (7) - v - a n + (- c + 1)j l - a j + (c + 1)g + a +--E 165 + +--S 166 +t8:=-c*j*n+c*j-c*n+j*n +--R +--R +--R (8) ((- c + 1)j - c)n + c j +--E 166 + +--S 167 +t9:=c*m*p-l*n*p+a*l+l*m+a*p-l*p+c-n-2 +--R +--R +--R (9) (- l n + c m - l + a)p - n + l m + a l + c - 2 +--E 167 + +--S 168 +t10:=-n^2*p+c*l+c*m+2*m*n+c*p-c*h+m +--R +--R +--R 2 +--R (10) (- n + c)p + 2m n + (c + 1)m + c l - c h +--E 168 + +--S 169 +t11:=-l*m*h+a*c+2*c*m-l*n+m*n-c*g-a+p+h +--R +--R +--R (11) p + (m - l)n + (- h l + 2c)m + h - c g + a c - a +--E 169 + +--S 170 +t12:=-c*l*m-c*m^2+a*m*n-m^2*n+c*m*v-m*n*h+c^2-c*j-n^2+n +--R +--R +--R 2 2 2 2 +--R (12) c m v - n + (- m + (- h + a)m + 1)n - c m - c l m - c j + c +--E 170 + +--S 171 +t13:=a^2*l-a*l^2+c*n*p-l*m*g-c*l-a*n+2*a-l+m-v+g +--R +--R +--R 2 2 +--R (13) - v + c n p - a n + (- g l + 1)m - a l + (- c + a - 1)l + g + 2a +--E 171 + +--S 172 +t14:=a*c*l-c*l^2+a*m*n-l*m*n-m*n*g+c*l*h-m^2+c*n-n^2+m*v+2*c +--R +--R +--R 2 2 2 +--R (14) m v - n + ((- l - g + a)m + c)n - m - c l + (c h + a c)l + 2c +--E 172 + +--S 173 +t15:=-j*l*m-l*n*v+c*l*g+a*l+c*n+n^2+j+1 +--R +--R +--R 2 +--R (15) - l n v + n + c n - j l m + (c g + a)l + j + 1 +--E 173 + +--S 174 +t16:=c*j*l-c*l*n-j*m*n+a*n^2-m*n^2+c*n*v-n^2*v+n*v +--R +--R +--R 2 2 +--R (16) (- n + (c + 1)n)v + (- m + a)n + (- j m - c l)n + c j l +--E 174 + +--S 175 +t17:=-j*l*p+c*m*p+n*p*h-a*l-a*m+a*g-p*g+a*h+c-j+n-1 +--R +--R +--R (17) (h n + c m - j l - g)p + n - a m - a l - j + a h + a g + c - 1 +--E 175 + +--S 176 +t18:=-j*n*p+l*m*h-c*l+j*m+c*g+n*h +--R +--R +--R (18) - j n p + h n + (h l + j)m - c l + c g +--E 176 + +--S 177 +t19:=l^2*h-l*h^2+a*c-c*l-j*l+c*m+j*m-n*g+c*h+n*h-a-l-g+h +--R +--R +--R (19) +--R 2 2 +--R (h - g)n + (j + c)m + h l + (- j - h - c - 1)l + (c + 1)h - g + a c - a +--E 177 + +--S 178 +t20:=a*j*m-j*m^2+c*m*v-c*m*g-c*m*h+l*n*h+n*v*h-n*h^2+c^2-c*n-2*j*n +--R +--R +--R 2 2 2 +--R (20) (h n + c m)v + (h l - 2j - h - c)n - j m + (a j - c h - c g)m + c +--E 178 + +--S 179 +t21:=j*n*p-l*g*h-j*l-n*g-m+h +--R +--R +--R (21) j n p - g n - m + (- j - g h)l + h +--E 179 + +--S 180 +t22:=j*l^2-j*l*v-a*n*g+n*g^2-j*l*h+2*j*n+l*g+m*g-v*g+j-1 +--R +--R +--R 2 2 +--R (22) (- j l - g)v + (2j + g - a g)n + g m + j l + (- h j + g)l + j - 1 +--E 180 + +--S 181 +t23:=j*l*n-j*m*n-c*n*g+j*n*g-j*n*h+j*m +--R +--R +--R (23) (- j m + j l + (- h + g)j - c g)n + j m +--E 181 + +--S 182 +t24:=-a^2*p+a*l*p+m^2*p-l*p*v+3*a+2*m-v +--R +--R +--R 2 2 +--R (24) (- l p - 1)v + (m + a l - a )p + 2m + 3a +--E 182 + +--S 183 +t25:=-a*c*p+c*l*p-n*p*v+n*p*h-a*m+l*m+m*v-m*h+2*c+2*n +--R +--R +--R (25) (- n p + m)v + (h n + c l - a c)p + 2n + (l - h - a)m + 2c +--E 183 + +--S 184 +t26:=-a*c*p+n*p*g+l^2-a*m-l*m+m^2+l*p-l*v+m*v-m*g-l*h-p*h+c+n +--R +--R +--R (26) +--R 2 2 +--R (m - l)v + (g n + l - h - a c)p + n + m + (- l - g - a)m + l - h l + c +--E 184 + +--S 185 +t27:=-c^2*p+j*n*p-2*c*m-j*m+l*n-m*n-n*h +--R +--R +--R 2 +--R (27) (j n - c )p + (- m + l - h)n + (- j - 2c)m +--E 185 + +--S 186 +t28:=m*n*p+n*p*v-a*l-l*m-a*p-l*v-l*g-p*g +--R +--R +--R (28) (n p - l)v + (m n - g - a)p - l m + (- g - a)l +--E 186 + +--S 187 +t29:=l*m*h-c*l-c*p-n*g+n*h-m +--R +--R +--R (29) - c p + (h - g)n + (h l - 1)m - c l +--E 187 + +--S 188 +t30:=l^2*v-l*v^2+l*m*g-j*l-a*n-l*n+m*n-j*p+2*n*v+n*g-v +--R +--R +--R 2 2 +--R (30) - l v + (2n + l - 1)v - j p + (m - l + g - a)n + g l m - j l +--E 188 + +--S 189 +t31:=j*l*m+l*n*v-c*n-n^2 +--R +--R +--R 2 +--R (31) l n v - n - c n + j l m +--E 189 + +@ +\section{Hietarinta2} +Variables $a,b,c,j,k,l,m,n,p,v$ +<<*>>= +)clear all + +--S 190 +t1:=-a*b*k+a*c*k+b*k*l-c*k*l-b^2*p+c^2*p+b*k +--R +--R +--R 2 2 +--R (1) (c - b )p + (- c + b)k l + (a c + (- a + 1)b)k +--E 190 + +--S 191 +t2:=-c^2*k+a*c*l+b*l*m-c*k*n+a*c+c^2+b*m +--R +--R +--R 2 2 +--R (2) - c k n + (b l + b)m + a c l - c k + c + a c +--E 191 + +--S 192 +t3:=a^2*b-a^2*c+2*b^2*k-2*c^2*k-a*b*l+a*c*l+b*l^2-c*l^2-a*b*m+a*c*m_ +-a*b-b^2+c^2+b*l-c*l +--R +--R +--R (3) +--R 2 2 2 2 +--R (a c - a b)m + (- c + b)l + ((a - 1)c + (- a + 1)b)l + (- 2c + 2b )k + c +--R + +--R 2 2 2 +--R - a c - b + (a - a)b +--E 192 + +--S 193 +t4:=-a*c^2+a*c*j-b*c*m-c^2*m+a*c*n+b*l*n-c*l*n+c^2*v+b*n-c*n +--R +--R +--R 2 2 2 +--R (4) c v + ((- c + b)l + (a - 1)c + b)n + (- c - b c)m + a c j - a c +--E 193 + +--S 194 +t5:=b^2*k+b*j*k-a*b*l-c*l*m-b^2 +--R +--R +--R 2 2 +--R (5) - c l m - a b l + (b j + b )k - b +--E 194 + +--S 195 +t6:=b*j*m-c*m*n+b*c +--R +--R +--R (6) - c m n + b j m + b c +--E 195 + +--S 196 +t7:=a*b^2-a*b*j+b*j*l-c*j*l+b^2*m+b*c*m-a*b*n-b^2*v +--R +--R +--R 2 2 2 +--R (7) - b v - a b n + (b c + b )m + (- c + b)j l - a b j + a b +--E 196 + +--S 197 +t8:=b*c*j-b*c*n+b*j*n-c*j*n +--R +--R +--R (8) ((- c + b)j - b c)n + b c j +--E 197 + +--S 198 +t9:=-2*b*k^2+c*k^2-a*k*l-k*l*m-k^2*n+a*b*p-b*l*p+c*m*p-l*n*p+b*k +--R +--R +--R 2 2 +--R (9) (- l n + c m - b l + a b)p - k n - k l m - a k l + (c - 2b)k + b k +--E 198 + +--S 199 +t10:=-b*k*m-c*k*m-2*k*m*n+b*c*p-n^2*p+c*k+b*m+c*m +--R +--R +--R 2 +--R (10) (- n + b c)p - 2k m n + ((- c - b)k + c + b)m + c k +--E 199 + +--S 200 +t11:=a*b*k-a*c*k-b*k*l-c*k*m-l^2*m+k*l*n-k*m*n+b^2*p-b*k+c*m-l*m-l*n +--R +--R +--R (11) +--R 2 2 +--R b p + (- k m + (k - 1)l)n + (- l - l - c k + c)m - b k l +--R + +--R (- a c + (a - 1)b)k +--E 200 + +--S 201 +t12:=-c^2*k+c*j*k-c*l*m-c*m^2-b*k*n+a*m*n-l*m*n-m^2*n+k*n^2+c*m*v+b*n-m*n-n^2 +--R +--R +--R (12) +--R 2 2 2 +--R c m v + (k - 1)n + (- m + (- l + a - 1)m - b k + b)n - c m - c l m +--R + +--R 2 +--R (c j - c )k +--E 201 + +--S 202 +t13:=-2*a*b*k+a^2*l+b*k*l+c*k*l-a*l^2-2*b*k*m-l*m^2+a*k*n+c*n*p_ + +b*k*v+a*b-b*l-l*n +--R +--R +--R (13) +--R 2 2 2 +--R b k v + c n p + (- l + a k)n - l m - 2b k m - a l + ((c + b)k - b + a )l +--R + +--R - 2a b k + a b +--E 202 + +--S 203 +t14:=-2*b*c*k+a*c*l-b*m^2-c*k*n+a*m*n-l*m*n-m^2*n+k*n^2+b*m*v_ + +b*c+c*l+c*n-n^2 +--R +--R +--R (14) +--R 2 2 2 +--R b m v + (k - 1)n + (- m + (- l + a)m - c k + c)n - b m + (a + 1)c l +--R + +--R - 2b c k + b c +--E 203 + +--S 204 +t15:=-b^2*k-b*j*k+a*b*l+c*l*m-j*l*m-c*k*n-k*n^2-l*n*v+b^2+c*n +--R +--R +--R (15) +--R 2 2 2 +--R - l n v - k n + (- c k + c)n + (- j + c)l m + a b l + (- b j - b )k + b +--E 204 + +--S 205 +t16:=c*j*l-c*l*n-j*m*n+a*n^2-m*n^2+b*n*v+c*n*v-n^2*v +--R +--R +--R 2 2 +--R (16) (- n + (c + b)n)v + (- m + a)n + (- j m - c l)n + c j l +--E 205 + +--S 206 +t17:=-b*k^2+c*k^2-j*k^2+k^2*n-j*l*p-b*m*p+c*m*p+l*n*p-a*k+n*p +--R +--R +--R 2 2 +--R (17) ((l + 1)n + (c - b)m - j l)p + k n + (- j + c - b)k - a k +--E 206 + +--S 207 +t18:=c*l*k-c*k*m-j*k*m+l^2*m-k*l*n-j*n*p+c*m+l*m-k*n+l*n+n +--R +--R +--R 2 +--R (18) - j n p + ((- k + 1)l - k + 1)n + (l + l + (- j - c)k + c)m + c k l +--E 207 + +--S 208 +t19:=a*b*k-a*c*k+j*k*l+b*k*m-c*k*m-j*k*m-k*l*n+k*m*n_ + -b*k-c*k-j*l-l^2-b*m+c*m-k*n+l*n-l+n +--R +--R +--R (19) +--R 2 +--R (k m + (- k + 1)l - k + 1)n + ((- j - c + b)k + c - b)m - l +--R + +--R (j k - j - 1)l + ((- a - 1)c + (a - 1)b)k +--E 208 + +--S 209 +t20:=-c^2*k+a*j*m-c*l*m-c*m^2-j*m^2+c*k*n+2*j*k*n+c*m*v+l*n*v_ + -c*m-j*n-l*n+n*v-n +--R +--R +--R (20) +--R 2 +--R ((l + 1)n + c m)v + (- l + (2j + c)k - j - 1)n + (- j - c)m +--R + +--R 2 +--R (- c l + a j - c)m - c k +--E 209 + +--S 210 +t21:=-b*k*l+j*k*l+b*k*m-l^2*m+k*m*n+j*n*p-b*k-j*l-b*m-l*m +--R +--R +--R 2 +--R (21) j n p + k m n + (- l - l + b k - b)m + ((j - b)k - j)l - b k +--E 210 + +--S 211 +t22:=b^2*k-b*j*k+b*l*m+b*m^2-2*j*k*n-a*m*n+m^2*n-j*l*v-b*m*v-j*l+j*n +--R +--R +--R (22) +--R 2 2 2 +--R (- b m - j l)v + (m - a m - 2j k + j)n + b m + b l m - j l + (- b j + b )k +--E 211 + +--S 212 +t23:=b*j*m-c*m*n-j*n +--R +--R +--R (23) (- c m - j)n + b j m +--E 212 + +--S 213 +t24:=3*a*k^2+2*k^2*m-a^2*p+a*l*p+m^2*p-k^2*v-l*p*v-2*a*k-b*p+n*p +--R +--R +--R 2 2 2 2 2 +--R (24) (- l p - k )v + (n + m + a l - b - a )p + 2k m + 3a k - 2a k +--E 213 + +--S 214 +t25:=2*c*k^2+a*k*m+2*k^2*n-a*c*p+c*l*p+l*n*p-k*m*v-n*p*v-c*k-a*m_ + +k*m+l*m+m^2-3*k*n+n*p+n +--R +--R +--R (25) +--R 2 2 +--R (- n p - k m)v + ((l + 1)n + c l - a c)p + (2k - 3k + 1)n + m +--R + +--R 2 +--R (l + (a + 1)k - a)m + 2c k - c k +--E 214 + +--S 215 +t26:=c*k^2+a*k*m+k*l*m+k^2*n-a*c*p+m*n*p+k*l*v-k*m*v_ + +2*b*k-c*k+k*l-a*m+m^2-2*k*n-b*p-l*v-b-l+n +--R +--R +--R (26) +--R 2 2 +--R (- k m + (k - 1)l)v + (m n - a c - b)p + (k - 2k + 1)n + m +--R + +--R 2 +--R (k l + a k - a)m + (k - 1)l + c k + (- c + 2b)k - b +--E 215 + +--S 216 +t27:=2*c*k*m+j*k*m+k*m*n-c^2*p+j*n*p-2*c*m+k*n-n +--R +--R +--R 2 +--R (27) (j n - c )p + (k m + k - 1)n + ((j + 2c)k - 2c)m +--E 216 + +--S 217 +t28:=a*k*l+2*k*l*m-a*b*p-b*m*p+m*n*p+k*l*v+n*p*v+b*k-l*m+k*n-l*v-b +--R +--R +--R (28) +--R (n p + (k - 1)l)v + (m n - b m - a b)p + k n + (2k - 1)l m + a k l + b k - b +--E 217 + +@ +\section{Hietarinta3} +Variables $x,y,z,t,u,v,w,a$ +<<*>>= +)clear all + +--S 218 +t1:=-x^2+y^2 +--R +--R +--R 2 2 +--R (1) y - x +--E 218 + +--S 219 +t2:=x*u*v+y*u*a-x-w +--R +--R +--R (2) a u y + (u v - 1)x - w +--E 219 + +--S 220 +t3:=x*u^2-y*u^2+y*z*a-x*u*a+y*u*a-x*v*a+x*a^2-y*a^2 +--R +--R +--R 2 2 2 2 +--R (3) a y z + (- u + a u - a )y + (- a v + u - a u + a )x +--E 220 + +--S 221 +t4:=-x*y*v-y^2*v+x*u*w-y*u*w+y*t*a+y*w*a-v^2+a^2 +--R +--R +--R 2 2 2 +--R (4) - v y + (- v x + (- u + a)w + a t)y + u w x - v + a +--E 221 + +--S 222 +t5:=-y*z*u-x*u*a+y+t +--R +--R +--R (5) - u y z + y - a u x + t +--E 222 + +--S 223 +t6:=x*y*z-x*y*v+x*t*v-y*z*w+z*u-u*v-z*a-v*a +--R +--R +--R (6) ((x - w)y + u - a)z - v x y + t v x + (- u - a)v +--E 223 + +--S 224 +t7:=x^2*z+x*y*z+x*t*u-y*t*u-x*t*a-x*w*a+z^2-a^2 +--R +--R +--R 2 2 2 +--R (7) z + (x y + x )z - t u y + (- a w + t u - a t)x - a +--E 224 + +--S 225 +t8:=x*y*t-x*y*w+x*t*w-y*t*w+x*z+z*t-y*v-v*w +--R +--R +--R (8) (x + t)z + ((- w + t)x - t w - v)y + t w x - v w +--E 225 + +--S 226 +t9:=-x*u+y*v-u*w+x*a +--R +--R +--R (9) v y + (- u + a)x - u w +--E 226 + +--S 227 +t10:=x*y-w^2 +--R +--R +--R 2 +--R (10) x y - w +--E 227 + +--S 228 +t11:=-u^2*v+x^2+z +--R +--R +--R 2 2 +--R (11) z + x - u v +--E 228 + +--S 229 +t12:=-y*u*v-y*v^2-u*v*w-v^2*w+y*v*a+u*w*a+x+t +--R +--R +--R 2 2 +--R (12) (- v + (- u + a)v)y + x + (- v - u v + a u)w + t +--E 229 + +--S 230 +t13:=-z*u*v-u^2*a+u*a^2+y*w+a +--R +--R +--R 2 2 +--R (13) - u v z + w y - a u + a u + a +--E 230 + +--S 231 +t14:=-x*v^2-z*v*w-u*v*w+y*u*a+x*v*a+v*w*a +--R +--R +--R 2 +--R (14) - v w z + a u y + (- v + a v)x + (- u + a)v w +--E 231 + +--S 232 +t15:=y*z*u-t*u*v+x*u*a-u*w*a +--R +--R +--R (15) u y z + a u x - a u w - t u v +--E 232 + +--S 233 +t16:=y*t*u-y*u*w-t*v*w-v*w^2+x*w*a+y*w*a-v^2+a^2 +--R +--R +--R 2 2 2 +--R (16) ((- u + a)w + t u)y + a w x - v w - t v w - v + a +--E 233 + +--S 234 +t17:=-x*z-t*u+y*v+u*w +--R +--R +--R (17) - x z + v y + u w - t u +--E 234 + +--S 235 +t18:=u^2*v-t*w-z +--R +--R +--R 2 +--R (18) - z - t w + u v +--E 235 + +--S 236 +t19:=-y*z*v-y*u*v-t*v^2+y*v*a+t*v*a+u*w*a +--R +--R +--R 2 +--R (19) - v y z + (- u + a)v y + a u w - t v + a t v +--E 236 + +--S 237 +t20:=-z*u^2+t*w+v +--R +--R +--R 2 +--R (20) - u z + t w + v +--E 237 + +--S 238 +t21:=x*z*u+x*z*v+z^2*w-x*z*a-t*u*a-z*w*a +--R +--R +--R 2 +--R (21) w z + ((v + u - a)x - a w)z - a t u +--E 238 + +--S 239 +t22:=x*t*v-y*z*w+z*t*w-t*v*w+z*u-u*v-z*a+v*a +--R +--R +--R (22) (- w y + t w + u - a)z + t v x - t v w + (- u + a)v +--E 239 + +--S 240 +t23:=v^2-a^2 +--R +--R +--R 2 2 +--R (23) v - a +--E 240 + +--S 241 +t24:=y*u+u*w-y*a-w*a +--R +--R +--R (24) (u - a)y + (u - a)w +--E 241 + +--S 242 +t25:=z*w-y*a +--R +--R +--R (25) w z - a y +--E 242 + +--S 243 +t26:=-y^2+t*w +--R +--R +--R 2 +--R (26) - y + t w +--E 243 + +--S 244 +t27:=-x*z+v*w-x*a+w*a +--R +--R +--R (27) - x z - a x + (v + a)w +--E 244 + +--S 245 +t28:=u^2*v-x*y-z +--R +--R +--R 2 +--R (28) - z - x y + u v +--E 245 + +--S 246 +t29:=z*u*v+u^2*a-u*a^2-x*t-a +--R +--R +--R 2 2 +--R (29) u v z - t x + a u - a u - a +--E 246 + +--S 247 +t30:=t*u*v+u*w*a-y-t +--R +--R +--R (30) - y + a u w + t u v - t +--E 247 + +--S 248 +t31:=-x*z+y*u+t*u-y*a +--R +--R +--R (31) - x z + (u - a)y + t u +--E 248 + +@ +\section{Hunecke} +Variables $x,y,z,t,u$ +<<*>>= +)clear all + +--S 249 +t1:=-x^5-y^5-z^5+5*x*y*z*t*u-u^5 +--R +--R +--R 5 5 5 5 +--R (1) - z + 5t u x y z - y - x - u +--E 249 + +--S 250 +t2:=x*y^3*z+y*z^3*t+x^3*y*u+z*t^3*u+z*t*u^3 +--R +--R +--R 3 3 3 3 3 +--R (2) t y z + (x y + t u + t u)z + u x y +--E 250 + +--S 251 +t3:=x^2*y*z^2+y^2*z*t^2+x^2*t^2*u+x*y^2*u^2+z^2*t*u^2 +--R +--R +--R 2 2 2 2 2 2 2 2 2 +--R (3) (x y + t u )z + t y z + u x y + t u x +--E 251 + +--S 252 +t4:=x*y*z^5-y^4*z^2*t-2*x^2*y^2*z*t*u+x*z^3*t^2*u-x^4*t*u^2_ + +y*z*t^2*u^3+x*y*u^5 +--R +--R +--R 5 2 3 4 2 2 2 2 3 5 2 4 +--R (4) x y z + t u x z - t y z + (- 2t u x y + t u y)z + u x y - t u x +--E 252 + +--S 253 +t5:=x*y^2*z^4-y^5*z*t-x^2*y^3*t*u+2*x*y*z^2*t^2*u+x*t^4*u^2_ + -x^2*y*z*u^3-z*t*u^5 +--R +--R +--R 2 4 2 2 5 3 2 5 2 3 4 2 +--R (5) x y z + 2t u x y z + (- t y - u x y - t u )z - t u x y + t u x +--E 253 + +--S 254 +t6:=x^3*y^2*t-y*z^2*t^4+x*y^2*z^3*u-y^5*t*u-t^6*u+3*x*y*z*t^2*u^2_ + -x^2*y*u^4-t*u^6 +--R +--R +--R 2 3 4 2 2 2 5 3 2 4 2 6 6 +--R (6) u x y z - t y z + 3t u x y z - t u y + t x y - u x y - t u - t u +--E 254 + +--S 255 +t7:=x^4*y^2*z-x*y*z^2*t^3-x*y^5*u-y^3*z^2*t*u-x*t^5*u_ + +2*x^2*y*z*t*u^2+z*t^2*u^4 +--R +--R +--R 3 3 2 4 2 2 2 2 4 5 5 +--R (7) (- t u y - t x y)z + (x y + 2t u x y + t u )z - u x y - t u x +--E 255 + +--S 256 +t8:=y^6*z+y*z^6+x^2*y^4*u-3*x*y^2*z^2*t*u+z^4*t^2*u-x^3*z*t*u^2_ +-x*y*t^3*u^3+y*z*u^5 +--R +--R +--R 6 2 4 2 2 6 5 2 3 2 4 3 3 +--R (8) y z + t u z - 3t u x y z + (y + u y - t u x )z + u x y - t u x y +--E 256 + +@ +\section{Laconcelli} +Variables $a,b,c,d,e,f,g,h,k,l,m$ +<<*>>= +)clear all + +--S 257 +t1:=a+b+c+d+e+f+g+h-1 +--R +--R +--R (1) h + g + f + e + d + c + b + a - 1 +--E 257 + +--S 258 +t2:=-a^2*k-2*a*b*k-b^2*k-a*c*k-b*c*k-a*d*k-b*d*k-a*e*k_ + -b*e*k-c*e*k-d*e*k-a*f*k-b*f*k-c*f*k-d*f*k+a+b +--R +--R +--R (2) +--R 2 +--R (- d - c - b - a)f + (- d - c - b - a)e + (- b - a)d + (- b - a)c - b +--R + +--R 2 +--R - 2a b - a +--R * +--R k +--R + +--R b + a +--E 258 + +--S 259 +t3:=-a^2*l-a*b*l-a*c*l-a*d*l-a*e*l-b*e*l-c*e*l-d*e*l_ + +a^2+2*a*b+b^2+a*e+b*e+a*f+b*f +--R +--R +--R (3) +--R 2 2 +--R ((- d - c - b - a)e - a d - a c - a b - a )l + (b + a)f + (b + a)e + b +--R + +--R 2 +--R 2a b + a +--E 259 + +--S 260 +t4:=a+c+e+g-m +--R +--R +--R (4) - m + g + e + c + a +--E 260 + +@ +\section{Morgenstern: articulated system (3 rods)} +Variables $x,y,z,t,u,v,w,a,b$ +<<*>>= +)clear all + +--S 261 +t1:=-y*z+x*t +--R +--R +--R (1) - y z + t x +--E 261 + +--S 262 +t2:=-y*u+x*v+y-v +--R +--R +--R (2) (- u + 1)y + v x - v +--E 262 + +--S 263 +t3:=z^2+t^2-w^2 +--R +--R +--R 2 2 2 +--R (3) z - w + t +--E 263 + +--S 264 +t4:=u^2+v^2-a^2-2*u+1 +--R +--R +--R 2 2 2 +--R (4) v + u - 2u - a + 1 +--E 264 + +--S 265 +t5:=z^2+t^2-2*z*u+u^2-2*t*v+v^2-b^2 +--R +--R +--R 2 2 2 2 2 +--R (5) z - 2u z + v - 2t v + u + t - b +--E 265 + +@ +\section{Pavelle Macsyma} +Variables $x,y,z,t,u,v,w,a$ +<<*>>= +)clear all + +--S 266 +t1:=x*y+x*z+x*t-u^2 +--R +--R +--R 2 +--R (1) x z + x y + t x - u +--E 266 + +--S 267 +t2:=x*y+y*z+y*t-v^2 +--R +--R +--R 2 +--R (2) y z + (x + t)y - v +--E 267 + +--S 268 +t3:=x*z+y*z+z*t-w^2 +--R +--R +--R 2 +--R (3) (y + x + t)z - w +--E 268 + +--S 269 +t4:=x*t+y*t+z*t-a^2 +--R +--R +--R 2 +--R (4) t z + t y + t x - a +--E 269 + +@ +\section{Raksanyi} +Variables $x,y,z,t,u,v,w,a$ +<<*>>= +)clear all + +--S 270 +t1:=t+v-a +--R +--R +--R (1) v + t - a +--E 270 + +--S 271 +t2:=x+y+z+t-u-w-a +--R +--R +--R (2) z + y + x - w - u + t - a +--E 271 + +--S 272 +t3:=x*z+y*z+x*t+z*t-u*w-u*a-w*a +--R +--R +--R (3) (y + x + t)z + t x + (- u - a)w - a u +--E 272 + +--S 273 +t4:=x*z*t-u*w*a +--R +--R +--R (4) t x z - a u w +--E 273 + +@ +\section{Rose} +Variables $x,y,z$ +<<*>>= + +--S 274 +t1:=y^4-20/7*x^2 +--R +--R +--R 4 20 2 +--R (5) y - -- x +--R 7 +--E 274 + +--S 275 +t2:=x^2*z^4 + 7/10*x*z^4 + 7/48*z^4 - 50/27*x^2 - 35/27*x - 49/216 +--R +--R +--R 2 7 7 4 50 2 35 49 +--R (6) (x + -- x + --)z - -- x - -- x - --- +--R 10 48 27 27 216 +--E 275 + +--S 276 +t3:=3/5*x^6*y^2*z + x^5*y^3 + 3/7*x^5*y^2*z + 7/5*x^4*y^3_ + - 7/20*x^4*y*z^2 - 3/20*x^4*z^3 + 609/1000*x^3*y^3_ + + 63/200*x^3*y^2*z - 77/125*x^3*y*z^2 - 21/50*x^3*z^3_ + + 49/1250*x^2*y^3 + 147/2000*x^2*y^2*z - 23863/60000*x^2*y*z^2_ + - 91/400*x^2*z^3 - 27391/800000*x*y^3 + 4137/800000*x*y^2*z_ + - 1078/9375*x*y*z^2 - 5887/200000*x*z^3 - 1029/160000*y^3_ + - 24353/1920000*y*z^2 - 343/128000*z^3 +--R +--R +--R (7) +--R 3 4 21 3 91 2 5887 343 3 +--R (- -- x - -- x - --- x - ------ x - ------)z +--R 20 50 400 200000 128000 +--R + +--R 7 4 77 3 23863 2 1078 24353 2 +--R (- -- x - --- x - ----- x - ---- x - -------)y z +--R 20 125 60000 9375 1920000 +--R + +--R 3 6 3 5 63 3 147 2 4137 2 +--R (- x + - x + --- x + ---- x + ------ x)y z +--R 5 7 200 2000 800000 +--R + +--R 5 7 4 609 3 49 2 27391 1029 3 +--R (x + - x + ---- x + ---- x - ------ x - ------)y +--R 5 1000 1250 800000 160000 +--E 276 + +)spool +)lisp (bye) + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} D. Bini and B. Mourrain. Polynomials test suite. 1996. +http://www-sop.inria.fr/saga/POL +\end{thebibliography} +\end{document}