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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT -1 39 "Maximum value of k in all calculations." }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "maxk:=60:" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 256 32 "Latin homogeneous 2x3 rectangles" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT -1 51 "This is the raw data calculated by a simpl
e method:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "for k from 1 to maxk \+
do\n  w[k]:=0:\n  s[k]:=0:\n   for x[2] from 1 to (k-1) do \n    for x
[3] from 1 to (k-1) do \n     for x[4] from 1 to (k-1) do " }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 76 "       x[1]:=2*x[2]+2*x[3]-3*x[4]:\n       \+
if ( (x[1]>0) and (x[1]<k) ) then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "        cs:=x[1]+x[4]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "    \+
    x[5]:=cs-x[2]: \n        x[6]:=cs-x[3]:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "          if ( (x[5]>0) and (x[5]<k) and (x[6]>0) an
d (x[6]<k) ) then \n            w[k]:=w[k]+1:\n            s[k]:=s[k]+
1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "            if ( (x[1]=x[2])
 or (x[1]=x[3]) or (x[2]=x[3]) or (x[4]=x[5]) or (x[4]=x[6]) or (x[5]=
x[6]) or (x[1]=x[4]) or (x[2]=x[5]) or (x[3]=x[6]) ) then s[k]:=s[k]-1
: fi:\n          fi:\n        fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "     od:\n    od:\n   od:\n  print(k,w[k],s[k]):\nod:" }}{PAGEBK }
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"
\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#\"\"\"\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"\"$\"\"#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"%\"\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"
&\"#;\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'\"#N\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"\"(\"#a\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"\")\"#\"*\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*\"$G\"\"
#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#5\"$*=\"#[" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"#6\"$]#\"#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#7
\"$T$\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8\"$K%\"$o\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#9\"$f&\"$S#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#:\"$'o\"$7$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#;
\"$b)\"$?%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#<\"%C5\"$G&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"#=\"%T7\"$s'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"#>\"%e9\"$;)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#?\"%H<\"%35
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#@\"%+?\"%+7" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"#A\"%JB\"%S9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#
B\"%iE\"%!o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#C\"%fI\"%!)>" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#D\"%cM\"%!G#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#E\"%DR\"%SE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#F
\"%%R%\"%+I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#G\"%T\\\"%KM" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#H\"%)[&\"%kQ" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#I\"%>h\"%oV" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#J
\"%]n\"%s[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#K\"%ru\"%ga" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"#L\"%#>)\"%[g" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#M\"%4!*\"%?n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#N
\"%E)*\"%#R(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#O\"&X2\"\"%g\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#P\"&k;\"\"%G*)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#Q\"&\"p7\"%#z*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"
#R\"&=P\"\"&c1\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#S\"&f[\"\"&G;\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#T\"&+g\"\"&+E\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%\"#U\"&hs\"\"&!o8" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6%\"#V\"&A&=\"&gZ\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#W\"&4*>\"&g
f\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#X\"&'H@\"&gr\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%\"#Y\"&:G#\"&![=" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"#Z\"&MV#\"&+)>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#[\"&\"*f#
\"&_7#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#\\\"&[w#\"&/F#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"#]\"&\\%H\"&)GC" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#^\"&]7$\"&se#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#
_\"&,K$\"&+w#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#`\"&_^$\"&G$H" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#a\"&fs$\"&+7$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#b\"&m$R\"&sI$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#
c\"&N;%\"&+^$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#d\"&/R%\"&Gr$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#e\"&Tj%\"&7$R" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#f\"&y([\"&'\\T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"
#g\"&*Q^\"&[Q%" }}}{PAGEBK }{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "We ex
pect a quasipolynomial for w and a quasipolynomial for s, both of degr
ee 3.  We don't know the period; the following calculations are set up
 to do any desired period.  The variables: " }}{PARA 0 "" 0 "" {TEXT 
-1 39 "p = assumed period of quasipolynomial, " }}{PARA 0 "" 0 "" 
{TEXT -1 35 "r (1<=r<=p) = constituent residue, " }}{PARA 0 "" 0 "" 
{TEXT -1 35 "d = degree of polynomial, dp = d+1." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 45 "The first step sets up the period and degree." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=4;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"pG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "dp := \+
3+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dpG\"\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 105 "Arrays to hold the coefficients of the weak an
d strong polynomials.  \"coef\" is a temporary working array." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "coef:=array(1..dp); \nwcoeff:=array
(1..p,1..dp);\nscoeff:=array(1..p,1..dp);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%coefG-%&arrayG6$;\"\"\"\"\"%7\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'wcoeffG-%&arrayG6%;\"\"\"\"\"%F(7\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'scoeffG-%&arrayG6%;\"\"\"\"\"%F(7\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 335 "The following procedure w
ill generate all the p different weak polynomials and p different stro
ng polynomials, factor them, and test by substituting the next value o
f the argument, comparing to the raw data of  the surplus period that \+
was calculated in the first procedure).  The polynomials will be saved
 in \"wpoly[r]\" and \"spoly[r]\"." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "for r from 1 to p do" }}{PARA 0 "" 0 "" {TEXT -1 127 "The followin
g procedure will generate the matrix of values for  numbers mod r of t
he period for degree d with any coefficients." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "V2:=array(1..dp,1..dp):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "for n from 1 to dp do \n  for k from 1 to dp do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "    V2[k,n]:=(p*(k-1)+r)^(n-1):\n  \+
od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "print(V2);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
-1 27 "This part assumes degree 3." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "print([w[r],w[r+p],w[r+2*p],w[r+3*p]]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "coef:=linsolve(V2,[w[r],w[r+p],w[r+2*p],w[r+3*p]]); \+
" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for j from 1 to \+
dp do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  wcoeff[r,j]:=coef[j]:\n
od;\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " coef:=linsolve(V2,[s[r],s
[r+p],s[r+2*p],s[r+3*p]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for j
 from 1 to dp do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  scoeff[r,j]:
=coef[j]:\nod;\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "wpoly[r]:=wcoe
ff[r,4]*x^3+wcoeff[r,3]*x^2+wcoeff[r,2]*x+wcoeff[r,1]; subs(x=r+4*p,wp
oly[r]); factor(wpoly[r]);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "sp
oly[r]:=scoeff[r,4]*x^3+scoeff[r,3]*x^2+scoeff[r,2]*x+scoeff[r,1]; sub
s(x=r+4*p,spoly[r]); factor(spoly[r]);\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%&arrayG
6%;\"\"\"\"\"%F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7
&\"\"\"F(F(F(7&F(\"\"&\"#D\"$D\"7&F(\"\"*\"#\")\"$H(7&F(\"#8\"$p\"\"%(
>#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!\"#;\"$G\"\"$K%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7&#!\"\"\"\"%#\"\"$F+#!
\"$F+#\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#
7&#!#:\"\"##\"#T\"\"%!\"$#\"\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%&wpolyG6#\"\"\",**&#F'\"\"%F'*$)%\"xG\"\"$F'F'F'*&#F/F+F'*$)F.\"\"#
F'F'!\"\"*&#F/F+F'F.F'F'#F'F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%
C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\",&%\"xG\"\"\"F)F&
\"\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&spolyG6#\"\"\",**&#F'\"
\"%F'*$)%\"xG\"\"$F'F'F'*&F/F')F.\"\"#F'!\"\"*&#\"#TF+F'F.F'F'#\"#:F2F
3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$**\"\"%!\"\",&%\"xG\"\"\"F)F&F),&F(F)\"\"&F&F),&F(F)
\"\"'F&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%&arrayG6%;\"\"
\"\"\"%F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"
\"\"#\"\"%\"\")7&F(\"\"'\"#O\"$;#7&F(\"#5\"$+\"\"%+57&F(\"#9\"$'>\"%WF
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"#N\"$*=\"$f&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7&!\"\"#\"\"$\"\"##!\"$
\"\"%#\"\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#
7&!#7\"#6!\"$#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&wpoly
G6#\"\"#,**&#\"\"\"\"\"%F+*$)%\"xG\"\"$F+F+F+*&#F0F,F+*$)F/F'F+F+!\"\"
*&#F0F'F+F/F+F+F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%T7" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"%!\"\",&%\"xG\"\"\"F)F&F),(*$)F(\"\"
#F)F)*&F-F)F(F)F&F%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&spoly
G6#\"\"#,**&#\"\"\"\"\"%F+*$)%\"xG\"\"$F+F+F+*&F0F+)F/F'F+!\"\"*&\"#6F
+F/F+F+\"#7F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$s'" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$**\"\"%!\"\",&%\"xG\"\"\"\"\"'F&F),&F(F)\"\"#F&F
),&F(F)F%F&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%&arrayG6%;
\"\"\"\"\"%F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"
\"\"\"\"$\"\"*\"#F7&F(\"\"(\"#\\\"$V$7&F(\"#6\"$@\"\"%J87&F(\"#:\"$D#
\"%vL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"#\"#a\"$]#\"$'o" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7&#!\"\"\"\"%#\"\"
$F+#!\"$F+#\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vect
orG6#7&#!#@\"\"##\"#T\"\"%!\"$#\"\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%&wpolyG6#\"\"$,**&#\"\"\"\"\"%F+*$)%\"xGF'F+F+F+*&#F'F,F+*$)F
/\"\"#F+F+!\"\"*&#F'F,F+F/F+F+#F+F,F5" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"%e9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\",&%\"xG\"\"
\"F)F&\"\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&spolyG6#\"\"$,**&
#\"\"\"\"\"%F+*$)%\"xGF'F+F+F+*&F'F+)F/\"\"#F+!\"\"*&#\"#TF,F+F/F+F+#
\"#@F2F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$;)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$**\"\"%!\"\",&%\"xG\"\"\"\"\"#F&F),&F(F)\"\"$F&F),&F(
F)\"\"(F&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%&arrayG6%;\"
\"\"\"\"%F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"
\"\"\"%\"#;\"#k7&F(\"\")F+\"$7&7&F(\"#7\"$W\"\"%G<7&F(F*\"$c#\"%'4%" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"*\"#\"*\"$T$\"$b)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7&!\"\"#\"\"$\"\"##!\"$\"\"%
#\"\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7&!#7
\"#6!\"$#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&wpolyG6#\"
\"%,**&#\"\"\"F'F+*$)%\"xG\"\"$F+F+F+*&#F/F'F+*$)F.\"\"#F+F+!\"\"*&#F/
F4F+F.F+F+F+F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%H<" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*(\"\"%!\"\",&%\"xG\"\"\"F)F&F),(*$)F(\"\"#F)F
)*&F-F)F(F)F&F%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&spolyG6#
\"\"%,**&#\"\"\"F'F+*$)%\"xG\"\"$F+F+F+*&F/F+)F.\"\"#F+!\"\"*&\"#6F+F.
F+F+\"#7F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%35" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$**\"\"%!\"\",&%\"xG\"\"\"\"\"'F&F),&F(F)\"\"#F&F),&
F(F)F%F&F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for r from \+
1 to p do: r: spoly[r]: factor(spoly[r]): od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"
\"%F&*$)%\"xG\"\"$F&F&F&*&F+F&)F*\"\"#F&!\"\"*&#\"#TF'F&F*F&F&#\"#:F.F
/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%!\"\",&%\"xG\"\"\"F)F&F)
,&F(F)\"\"&F&F),&F(F)\"\"'F&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"%F&*$)%\"xG\"\"$
F&F&F&*&F+F&)F*\"\"#F&!\"\"*&\"#6F&F*F&F&\"#7F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$**\"\"%!\"\",&%\"xG\"\"\"\"\"'F&F),&F(F)\"\"#F&F),&F(
F)F%F&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,**&#\"\"\"\"\"%F&*$)%\"xG\"\"$F&F&F&*&F+F&)F*\"\"#F&
!\"\"*&#\"#TF'F&F*F&F&#\"#@F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*
*\"\"%!\"\",&%\"xG\"\"\"\"\"#F&F),&F(F)\"\"$F&F),&F(F)\"\"(F&F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,**&#\"\"\"\"\"%F&*$)%\"xG\"\"$F&F&F&*&F+F&)F*\"\"#F&!\"\"*&\"#6F&F
*F&F&\"#7F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%!\"\",&%\"xG\"
\"\"\"\"'F&F),&F(F)\"\"#F&F),&F(F)F%F&F)F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "r:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for k fr
om 1 to maxk do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "  r:=r+1: " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "  if (r>p) then r:=1: fi:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "  eval(spoly[r],x=k)-s[k]: " }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 
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" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
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11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
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\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" 
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11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
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\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
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11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 
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" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"&" }}{PARA 11 "" 1 "" 
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20 "6#>%\"rG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
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\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for
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1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"
\"\"\"\"%F&*$)%\"xG\"\"$F&F&F&*&#F+F'F&*$)F*\"\"#F&F&!\"\"*&#F+F'F&F*F
&F&#F&F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\",&%\"xG\"
\"\"F)F&\"\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"%F&*$)%\"xG\"\"$F&F&F&*&#F+F'F&*$
)F*\"\"#F&F&!\"\"*&#F+F0F&F*F&F&F&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,$*(\"\"%!\"\",&%\"xG\"\"\"F)F&F),(*$)F(\"\"#F)F)*&F-F)F(F)F&F%F)F)F
)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**&#\"\"\"\"\"%F&*$)%\"xG\"\"$F&F&F&*&#F+F'F&*$)F*\"\"
#F&F&!\"\"*&#F+F'F&F*F&F&#F&F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$
*&\"\"%!\"\",&%\"xG\"\"\"F)F&\"\"$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"%F&*$)%\"xG
\"\"$F&F&F&*&#F+F'F&*$)F*\"\"#F&F&!\"\"*&#F+F0F&F*F&F&F&F1" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"%!\"\",&%\"xG\"\"\"F)F&F),(*$)F(\"\"
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57 "r:=1;for k from 1 to maxk do \n  eval(wpoly[r],x=k)-w[k]: " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "  r:=3-r: \nod;\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" 
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11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
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11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
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11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
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11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" 
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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
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11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
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11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" 
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{XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "2 0 0" 32 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }