For Formula (19.125)  on page 217 of ChalkleyJan05.dvi
Verification when n = 1, 2, 3.

\(difference[n_]\  := \ 
      Module[\ \ {vect, \ m, \ M, \ F, \ Fprime, \ G, Gpri, difference}, 

        		vect[z_]\  = \ Table[\ \(f[i]\)[z], {i, 0, n}]; \ 
		
        \(m[i_, j_]\)[z_]\  := \ 
          Sum[\ Binomial[n - i, j - nu]*Binomial[i, nu]*
              a^\((n - i - j + nu)\)*b^\((j - nu)\)*c^\((i - nu)\)*d^nu, {nu, 
              0, j}]; 
		
        M[z_]\  = \ Table[\ \(m[i, j]\)[z], {i, 0, n}, {j, 0, n}]; 
		\ 
        Do[\ \ \(g[i]\)[z_]\  = \ 
            \(Flatten[\((vect[z].M[z])\)]\)[\([i + 1]\)], \ {i, 0, n}]; 
		
        F[Y_]\  = \ Sum[\ \(f[i]\)[z]*Y^i, {i, 0, n}\ ]; 
		
        Fprime[Y_]\  = \ \ Sum[\ \(f[i]'\)[z]*Y^i, {i, 0, n}\ ]; 
		
        G[Y_]\  = \ Sum[\ \(g[i]\)[z]*Y^i, {i, 0, n}]; 
		
        Gpri[Y_]\  = \ Sum[\ \(g[i]'\)[z]*Y^i, {i, 0, n}]; 
		
        difference\  = \ 
          Expand[\ \ \ 
            \((\ Resultant[G[Y], Gpri[Y], Y] - 
                \((a*d - b*c)\)^\((n^2)\)*Resultant[F[Y], Fprime[Y], Y])
              \)\ \ ]; 
		
        Print[\ "difference[", n, "] = ", \ difference]\ \ ]\)

\(ratio[n_]\  := \ 
      Module[\ \ {vect, \ m, \ M, \ F, \ Fprime, \ G, Gpri, ratio}, 
		
        vect[z_]\  = \ Table[\ \(f[i]\)[z], {i, 0, n}]; \ 
		
        \(m[i_, j_]\)[z_]\  := \ 
          Sum[\ Binomial[n - i, j - nu]*Binomial[i, nu]*
              a^\((n - i - j + nu)\)*b^\((j - nu)\)*c^\((i - nu)\)*d^nu, {nu, 
              0, j}]; 
		
        M[z_]\  = \ Table[\ \(m[i, j]\)[z], {i, 0, n}, {j, 0, n}]; 
		\ 
        Do[\ \ \(g[i]\)[z_]\  = \ 
            \(Flatten[\((vect[z].M[z])\)]\)[\([i + 1]\)], \ {i, 0, n}]; 
		
        F[Y_]\  = \ Sum[\ \(f[i]\)[z]*Y^i, {i, 0, n}\ ]; 
		
        Fprime[Y_]\  = \ \ Sum[\ \(f[i]'\)[z]*Y^i, {i, 0, n}\ ]; 
		
        G[Y_]\  = \ Sum[\ \(g[i]\)[z]*Y^i, {i, 0, n}]; 
		
        Gpri[Y_]\  = \ Sum[\ \(g[i]'\)[z]*Y^i, {i, 0, n}]; 
		
        ratio\  = \ 
          Factor[\((\ 
              Resultant[G[Y], Gpri[Y], Y]/Resultant[F[Y], Fprime[Y], Y])
              \)\ ]; 
		Print[\ "ratio[", n, "] = ", \ ratio]\ \ ]
        \)

Date[]

difference[1]

Date[]

ratio[1]

Date[]

difference[2]

Date[]

ratio[2]

Date[]

difference[3]

Date[]

ratio[3]

Date[]

The case n = 4 was run separately on a machine having 1024 MB of RAM and required about 5 hours.  

All of the other computations for ChalkleyJuly04.dvi were done on a machine having 512 MB of RAM and the following information:

System Information report written at: 06/13/04 23:17:14
System Name: YOUR-US67PI6LUV
[System Summary (C:\DOCUME~1\Owner\MYDOCU~1\Word\WO72BB~1\SYSTEM~1.NFO)]

Item	Value	
OS Name	Microsoft Windows XP Home Edition	
Version	5.1.2600 Service Pack 1 Build 2600	
OS Manufacturer	Microsoft Corporation	
System Name	YOUR-US67PI6LUV	
System Manufacturer	HP Pavilion 05	
System Model	P9852A-ABA 753N	
System Type	X86-based PC	
Processor	x86 Family 15 Model 2 Stepping 7 GenuineIntel ~2524 Mhz	
BIOS Version/Date	Phoenix Technologies, LTD VG31313, 9/11/2002	
SMBIOS Version	2.3	
Windows Directory	C:\WINDOWS	
System Directory	C:\WINDOWS\System32	
Boot Device	\Device\HarddiskVolume2	
Locale	United States	
Hardware Abstraction Layer	Version = "5.1.2600.1106 (xpsp1.020828-1920)"	
User Name	YOUR-US67PI6LUV\Owner	
Time Zone	Eastern Standard Time	
Total Physical Memory	512.00 MB	
Available Physical Memory	286.24 MB	
Total Virtual Memory	1.72 GB	
Available Virtual Memory	1.35 GB	
Page File Space	1.22 GB	
Page File	C:\pagefile.sys	

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