How to get a rational solution of the linear system efficiently?

Question

I got a large linear system p={Subscript[a, 1,24]>=0,1/2-Subscript[a, 1,24]-Subscript[a, 1,25]>=0,Subscript[a, 1,25]>=0,Subscript[a, 4,5]>=0,10033/37620-Subscript[a, 4,5]-Subscript[a, 4,18]>=0,Subscript[a, 4,18]>=0,Subscript[a, 5,4]>=0,Subscript[a, 5,8]>=0,887/1050-Subscript[a, 5,4]-Subscript[a, 5,8]-Subscript[a, 5,17]>=0,Subscript[a, 5,17]>=0,Subscript[a, 7,5]>=0,3/4-Subscript[a, 7,5]-Subscript[a, 7,13]>=0,Subscript[a, 7,13]>=0,Subscript[a, 10,10]>=0,25/124-Subscript[a, 10,10]-Subscript[a, 10,24]>=0,Subscript[a, 10,24]>=0,Subscript[a, 11,2]>=0,Subscript[a, 11,4]>=0,1-Subscript[a, 11,2]-Subscript[a, 11,4]-Subscript[a, 11,8]>=0,Subscript[a, 11,8]>=0,Subscript[a, 12,14]>=0,1-Subscript[a, 12,14]-Subscript[a, 12,17]>=0,Subscript[a, 12,17]>=0,Subscript[a, 13,2]>=0,6/7-Subscript[a, 13,2]-Subscript[a, 13,10]>=0,Subscript[a, 13,10]>=0,Subscript[a, 17,2]>=0,1321/1575-Subscript[a, 17,2]-Subscript[a, 17,14]>=0,Subscript[a, 17,14]>=0,Subscript[a, 19,4]>=0,Subscript[a, 19,8]>=0,Subscript[a, 19,18]>=0,1-Subscript[a, 19,4]-Subscript[a, 19,8]-Subscript[a, 19,18]-Subscript[a, 19,26]>=0,Subscript[a, 19,26]>=0,28-31 Subscript[a, 11,2]-33 Subscript[a, 13,2]-24 Subscript[a, 17,2]-16 Subscript[a, 23,2]>=0,4-5 Subscript[a, 11,2]-4 Subscript[a, 13,2]-4 Subscript[a, 17,2]-3 Subscript[a, 23,2]>=0,Subscript[a, 23,2]>=0,-14+15 Subscript[a, 11,2]+16 Subscript[a, 13,2]+13 Subscript[a, 17,2]+9 Subscript[a, 23,2]>=0,-17+20 Subscript[a, 11,2]+20 Subscript[a, 13,2]+14 Subscript[a, 17,2]+9 Subscript[a, 23,2]>=0,37614/40889-Subscript[a, 23,2]-Subscript[a, 23,5]>=0,Subscript[a, 23,5]>=0,Subscript[a, 24,4]>=0,Subscript[a, 24,13]>=0,95/132-Subscript[a, 24,4]-Subscript[a, 24,13]-Subscript[a, 24,26]>=0,Subscript[a, 24,26]>=0,Subscript[a, 25,18]>=0,3/4-Subscript[a, 25,18]-Subscript[a, 25,26]>=0,Subscript[a, 25,26]>=0,Subscript[a, 26,5]>=0,12/19-Subscript[a, 26,5]-Subscript[a, 26,17]>=0,Subscript[a, 26,17]>=0,Subscript[a, 27,26]>=0,1/2-Subscript[a, 27,26]-Subscript[a, 27,30]>=0,Subscript[a, 27,30]>=0,54-38 Subscript[a, 5,4]-31 Subscript[a, 11,4]-21 Subscript[a, 19,4]-18 Subscript[a, 24,4]-15 Subscript[a, 28,4]>=0,8-5 Subscript[a, 5,4]-5 Subscript[a, 11,4]-4 Subscript[a, 19,4]-3 Subscript[a, 24,4]-2 Subscript[a, 28,4]>=0,Subscript[a, 28,4]>=0,-26+19 Subscript[a, 5,4]+15 Subscript[a, 11,4]+12 Subscript[a, 19,4]+8 Subscript[a, 24,4]+6 Subscript[a, 28,4]>=0,-34+23 Subscript[a, 5,4]+20 Subscript[a, 11,4]+12 Subscript[a, 19,4]+12 Subscript[a, 24,4]+10 Subscript[a, 28,4]>=0,313/440-Subscript[a, 28,4]-Subscript[a, 28,13]>=0,Subscript[a, 28,13]>=0,601824/40889-38 Subscript[a, 5,8]-31 Subscript[a, 11,8]-21 Subscript[a, 19,8]-13 Subscript[a, 29,8]>=0,297637/122667-5 Subscript[a, 5,8]-5 Subscript[a, 11,8]-4 Subscript[a, 19,8]-2 Subscript[a, 29,8]>=0,Subscript[a, 29,8]>=0,-(933800/122667)+19 Subscript[a, 5,8]+15 Subscript[a, 11,8]+12 Subscript[a, 19,8]+5 Subscript[a, 29,8]>=0,-(1097356/122667)+23 Subscript[a, 5,8]+20 Subscript[a, 11,8]+12 Subscript[a, 19,8]+9 Subscript[a, 29,8]>=0,Subscript[a, 29,10]>=0,49/76-Subscript[a, 29,8]-Subscript[a, 29,10]-Subscript[a, 29,24]>=0,Subscript[a, 29,24]>=0,Subscript[a, 30,24]>=0,8/15-Subscript[a, 30,24]-Subscript[a, 30,25]>=0,Subscript[a, 30,25]>=0,Subscript[a, 31,13]>=0,-(10517/1260)+17 Subscript[a, 7,13]+8 Subscript[a, 24,13]+6 Subscript[a, 28,13]+5 Subscript[a, 31,13]>=0,Subscript[a, 31,22]>=0,1-Subscript[a, 31,13]-Subscript[a, 31,22]-Subscript[a, 31,26]>=0,Subscript[a, 31,26]>=0,Subscript[a, 32,25]>=0,1-Subscript[a, 32,25]-Subscript[a, 32,26]>=0,Subscript[a, 32,26]>=0,45-38 Subscript[a, 4,5]-33 Subscript[a, 7,5]-16 Subscript[a, 23,5]-18 Subscript[a, 26,5]-9 Subscript[a, 34,5]>=0,7-6 Subscript[a, 4,5]-5 Subscript[a, 7,5]-3 Subscript[a, 23,5]-2 Subscript[a, 26,5]-2 Subscript[a, 34,5]>=0,Subscript[a, 34,5]>=0,-23+19 Subscript[a, 4,5]+17 Subscript[a, 7,5]+9 Subscript[a, 23,5]+8 Subscript[a, 26,5]+4 Subscript[a, 34,5]>=0,-27+24 Subscript[a, 4,5]+20 Subscript[a, 7,5]+9 Subscript[a, 23,5]+11 Subscript[a, 26,5]+6 Subscript[a, 34,5]>=0,7/11-Subscript[a, 34,5]-Subscript[a, 34,14]>=0,Subscript[a, 34,14]>=0,Subscript[a, 36,10]>=0,1-Subscript[a, 36,10]-Subscript[a, 36,22]>=0,Subscript[a, 36,22]>=0,14-41 Subscript[a, 1,24]-29 Subscript[a, 10,24]-13 Subscript[a, 29,24]-12 Subscript[a, 30,24]-12 Subscript[a, 37,24]>=0,2-6 Subscript[a, 1,24]-5 Subscript[a, 10,24]-2 Subscript[a, 29,24]-2 Subscript[a, 30,24]-Subscript[a, 37,24]>=0,Subscript[a, 37,24]>=0,-6+21 Subscript[a, 1,24]+16 Subscript[a, 10,24]+5 Subscript[a, 29,24]+5 Subscript[a, 30,24]+5 Subscript[a, 37,24]>=0,-9+25 Subscript[a, 1,24]+17 Subscript[a, 10,24]+9 Subscript[a, 29,24]+8 Subscript[a, 30,24]+7 Subscript[a, 37,24]>=0,3347/4650-Subscript[a, 37,24]-Subscript[a, 37,30]>=0,Subscript[a, 37,30]>=0,79/180-Subscript[a, 42,22]>=0,37/28-2 Subscript[a, 31,22]-Subscript[a, 36,22]-Subscript[a, 42,22]>=0,Subscript[a, 42,22]>=0,23-29 Subscript[a, 10,10]-33 Subscript[a, 13,10]-13 Subscript[a, 29,10]-13 Subscript[a, 36,10]-3 Subscript[a, 43,10]>=0,3-5 Subscript[a, 10,10]-4 Subscript[a, 13,10]-2 Subscript[a, 29,10]-Subscript[a, 36,10]-Subscript[a, 43,10]>=0,Subscript[a, 43,10]>=0,-14+17 Subscript[a, 10,10]+20 Subscript[a, 13,10]+9 Subscript[a, 29,10]+8 Subscript[a, 36,10]+Subscript[a, 43,10]>=0,-11+16 Subscript[a, 10,10]+16 Subscript[a, 13,10]+5 Subscript[a, 29,10]+5 Subscript[a, 36,10]+2 Subscript[a, 43,10]>=0,471/19-30 Subscript[a, 12,14]-24 Subscript[a, 17,14]-9 Subscript[a, 34,14]-3 Subscript[a, 43,14]>=0,85/19-5 Subscript[a, 12,14]-4 Subscript[a, 17,14]-2 Subscript[a, 34,14]-Subscript[a, 43,14]>=0,Subscript[a, 43,14]>=0,-(283/19)+19 Subscript[a, 12,14]+14 Subscript[a, 17,14]+6 Subscript[a, 34,14]+Subscript[a, 43,14]>=0,-(245/19)+15 Subscript[a, 12,14]+13 Subscript[a, 17,14]+4 Subscript[a, 34,14]+2 Subscript[a, 43,14]>=0,36-21 Subscript[a, 19,26]-18 Subscript[a, 24,26]-20 Subscript[a, 25,26]-17 Subscript[a, 27,26]-9 Subscript[a, 31,26]-8 Subscript[a, 32,26]-3 Subscript[a, 43,26]>=0,6-4 Subscript[a, 19,26]-3 Subscript[a, 24,26]-2 Subscript[a, 25,26]-2 Subscript[a, 27,26]-2 Subscript[a, 31,26]-2 Subscript[a, 32,26]-Subscript[a, 43,26]>=0,11/12-Subscript[a, 43,10]-Subscript[a, 43,14]-Subscript[a, 43,26]>=0,Subscript[a, 43,26]>=0,-21+12 Subscript[a, 19,26]+12 Subscript[a, 24,26]+12 Subscript[a, 25,26]+11 Subscript[a, 27,26]+5 Subscript[a, 31,26]+4 Subscript[a, 32,26]+Subscript[a, 43,26]>=0,-18+12 Subscript[a, 19,26]+8 Subscript[a, 24,26]+9 Subscript[a, 25,26]+7 Subscript[a, 27,26]+5 Subscript[a, 31,26]+5 Subscript[a, 32,26]+2 Subscript[a, 43,26]>=0,Subscript[a, 44,17]>=0,-(2075/88)+23 Subscript[a, 5,17]+19 Subscript[a, 12,17]+11 Subscript[a, 26,17]+2 Subscript[a, 44,17]>=0,29/31-Subscript[a, 44,17]-Subscript[a, 44,25]>=0,Subscript[a, 44,25]>=0,-14+21 Subscript[a, 1,25]+5 Subscript[a, 30,25]+5 Subscript[a, 32,25]+Subscript[a, 44,25]>=0,17-38 Subscript[a, 4,18]-21 Subscript[a, 19,18]-20 Subscript[a, 25,18]-2 Subscript[a, 45,18]>=0,3-6 Subscript[a, 4,18]-4 Subscript[a, 19,18]-2 Subscript[a, 25,18]-Subscript[a, 45,18]>=0,Subscript[a, 45,18]>=0,-9+19 Subscript[a, 4,18]+12 Subscript[a, 19,18]+9 Subscript[a, 25,18]+Subscript[a, 45,18]>=0,-10+24 Subscript[a, 4,18]+12 Subscript[a, 19,18]+12 Subscript[a, 25,18]+Subscript[a, 45,18]>=0,194/19-17 Subscript[a, 27,30]-12 Subscript[a, 37,30]-2 Subscript[a, 45,30]>=0,29/19-2 Subscript[a, 27,30]-Subscript[a, 37,30]-Subscript[a, 45,30]>=0,1-Subscript[a, 45,18]-Subscript[a, 45,30]>=0,Subscript[a, 45,30]>=0,-(78/19)+7 Subscript[a, 27,30]+5 Subscript[a, 37,30]+Subscript[a, 45,30]>=0,-(116/19)+11 Subscript[a, 27,30]+7 Subscript[a, 37,30]+Subscript[a, 45,30]>=0,1769/1980-Subscript[a, 46,17]>=0,3383/88-38 Subscript[a, 5,17]-30 Subscript[a, 12,17]-18 Subscript[a, 26,17]-3 Subscript[a, 44,17]-Subscript[a, 46,17]>=0,1537/220-5 Subscript[a, 5,17]-5 Subscript[a, 12,17]-2 Subscript[a, 26,17]-Subscript[a, 44,17]-Subscript[a, 46,17]>=0,Subscript[a, 46,17]>=0,-(4263/220)+19 Subscript[a, 5,17]+15 Subscript[a, 12,17]+8 Subscript[a, 26,17]+Subscript[a, 44,17]+Subscript[a, 46,17]>=0,7367/420-33 Subscript[a, 7,13]-18 Subscript[a, 24,13]-15 Subscript[a, 28,13]-9 Subscript[a, 31,13]-Subscript[a, 47,13]>=0,992/315-5 Subscript[a, 7,13]-3 Subscript[a, 24,13]-2 Subscript[a, 28,13]-2 Subscript[a, 31,13]-Subscript[a, 47,13]>=0,Subscript[a, 47,13]>=0,-(27893/2520)+20 Subscript[a, 7,13]+12 Subscript[a, 24,13]+10 Subscript[a, 28,13]+5 Subscript[a, 31,13]+Subscript[a, 47,13]>=0,28-41 Subscript[a, 1,25]-12 Subscript[a, 30,25]-8 Subscript[a, 32,25]-3 Subscript[a, 44,25]-Subscript[a, 47,25]>=0,5-6 Subscript[a, 1,25]-2 Subscript[a, 30,25]-2 Subscript[a, 32,25]-Subscript[a, 44,25]-Subscript[a, 47,25]>=0,161/198-Subscript[a, 47,13]-Subscript[a, 47,25]>=0,Subscript[a, 47,25]>=0,-17+25 Subscript[a, 1,25]+8 Subscript[a, 30,25]+4 Subscript[a, 32,25]+2 Subscript[a, 44,25]+Subscript[a, 47,25]>=0,-(622550989463/10767709260)+6 Subscript[a, 1,24]+6 Subscript[a, 1,25]+6 Subscript[a, 4,5]+6 Subscript[a, 4,18]+5 Subscript[a, 5,4]+5 Subscript[a, 5,8]+5 Subscript[a, 5,17]+5 Subscript[a, 7,5]+5 Subscript[a, 7,13]+5 Subscript[a, 10,10]+5 Subscript[a, 10,24]+5 Subscript[a, 11,2]+5 Subscript[a, 11,4]+5 Subscript[a, 11,8]+5 Subscript[a, 12,14]+5 Subscript[a, 12,17]+4 Subscript[a, 13,2]+4 Subscript[a, 13,10]+4 Subscript[a, 17,2]+4 Subscript[a, 17,14]+4 Subscript[a, 19,4]+4 Subscript[a, 19,8]+4 Subscript[a, 19,18]+4 Subscript[a, 19,26]+3 Subscript[a, 23,2]+3 Subscript[a, 23,5]+3 Subscript[a, 24,4]+3 Subscript[a, 24,13]+3 Subscript[a, 24,26]+2 Subscript[a, 25,18]+2 Subscript[a, 25,26]+2 Subscript[a, 26,5]+2 Subscript[a, 26,17]+2 Subscript[a, 27,26]+2 Subscript[a, 27,30]+2 Subscript[a, 28,4]+2 Subscript[a, 28,13]+2 Subscript[a, 29,8]+2 Subscript[a, 29,10]+2 Subscript[a, 29,24]+2 Subscript[a, 30,24]+2 Subscript[a, 30,25]+2 Subscript[a, 31,13]+2 Subscript[a, 31,22]+2 Subscript[a, 31,26]+2 Subscript[a, 32,25]+2 Subscript[a, 32,26]+2 Subscript[a, 34,5]+2 Subscript[a, 34,14]+Subscript[a, 36,10]+Subscript[a, 36,22]+Subscript[a, 37,24]+Subscript[a, 37,30]+Subscript[a, 42,22]+Subscript[a, 43,10]+Subscript[a, 43,14]+Subscript[a, 43,26]+Subscript[a, 44,17]+Subscript[a, 44,25]+Subscript[a, 45,18]+Subscript[a, 45,30]+Subscript[a, 46,17]+Subscript[a, 47,13]+Subscript[a, 47,25]>=0,421/28-9 Subscript[a, 31,22]-13 Subscript[a, 36,22]-9 Subscript[a, 42,22]-3 Subscript[a, 49,22]>=0,13/93-Subscript[a, 49,22]>=0,1927120621541/10767709260-21 Subscript[a, 1,24]-21 Subscript[a, 1,25]-19 Subscript[a, 4,5]-19 Subscript[a, 4,18]-19 Subscript[a, 5,4]-19 Subscript[a, 5,8]-19 Subscript[a, 5,17]-17 Subscript[a, 7,5]-17 Subscript[a, 7,13]-16 Subscript[a, 10,10]-16 Subscript[a, 10,24]-15 Subscript[a, 11,2]-15 Subscript[a, 11,4]-15 Subscript[a, 11,8]-15 Subscript[a, 12,14]-15 Subscript[a, 12,17]-16 Subscript[a, 13,2]-16 Subscript[a, 13,10]-13 Subscript[a, 17,2]-13 Subscript[a, 17,14]-12 Subscript[a, 19,4]-12 Subscript[a, 19,8]-12 Subscript[a, 19,18]-12 Subscript[a, 19,26]-9 Subscript[a, 23,2]-9 Subscript[a, 23,5]-8 Subscript[a, 24,4]-8 Subscript[a, 24,13]-8 Subscript[a, 24,26]-9 Subscript[a, 25,18]-9 Subscript[a, 25,26]-8 Subscript[a, 26,5]-8 Subscript[a, 26,17]-7 Subscript[a, 27,26]-7 Subscript[a, 27,30]-6 Subscript[a, 28,4]-6 Subscript[a, 28,13]-5 Subscript[a, 29,8]-5 Subscript[a, 29,10]-5 Subscript[a, 29,24]-5 Subscript[a, 30,24]-5 Subscript[a, 30,25]-5 Subscript[a, 31,13]-5 Subscript[a, 31,22]-5 Subscript[a, 31,26]-5 Subscript[a, 32,25]-5 Subscript[a, 32,26]-4 Subscript[a, 34,5]-4 Subscript[a, 34,14]-5 Subscript[a, 36,10]-5 Subscript[a, 36,22]-5 Subscript[a, 37,24]-5 Subscript[a, 37,30]-3 Subscript[a, 42,22]-2 Subscript[a, 43,10]-2 Subscript[a, 43,14]-2 Subscript[a, 43,26]-Subscript[a, 44,17]-Subscript[a, 44,25]-Subscript[a, 45,18]-Subscript[a, 45,30]-Subscript[a, 46,17]-Subscript[a, 49,22]>=0,1197471743731/5383854630-25 Subscript[a, 1,24]-25 Subscript[a, 1,25]-24 Subscript[a, 4,5]-24 Subscript[a, 4,18]-23 Subscript[a, 5,4]-23 Subscript[a, 5,8]-23 Subscript[a, 5,17]-20 Subscript[a, 7,5]-20 Subscript[a, 7,13]-17 Subscript[a, 10,10]-17 Subscript[a, 10,24]-20 Subscript[a, 11,2]-20 Subscript[a, 11,4]-20 Subscript[a, 11,8]-19 Subscript[a, 12,14]-19 Subscript[a, 12,17]-20 Subscript[a, 13,2]-20 Subscript[a, 13,10]-14 Subscript[a, 17,2]-14 Subscript[a, 17,14]-12 Subscript[a, 19,4]-12 Subscript[a, 19,8]-12 Subscript[a, 19,18]-12 Subscript[a, 19,26]-9 Subscript[a, 23,2]-9 Subscript[a, 23,5]-12 Subscript[a, 24,4]-12 Subscript[a, 24,13]-12 Subscript[a, 24,26]-12 Subscript[a, 25,18]-12 Subscript[a, 25,26]-11 Subscript[a, 26,5]-11 Subscript[a, 26,17]-11 Subscript[a, 27,26]-11 Subscript[a, 27,30]-10 Subscript[a, 28,4]-10 Subscript[a, 28,13]-9 Subscript[a, 29,8]-9 Subscript[a, 29,10]-9 Subscript[a, 29,24]-8 Subscript[a, 30,24]-8 Subscript[a, 30,25]-5 Subscript[a, 31,13]-5 Subscript[a, 31,22]-5 Subscript[a, 31,26]-4 Subscript[a, 32,25]-4 Subscript[a, 32,26]-6 Subscript[a, 34,5]-6 Subscript[a, 34,14]-8 Subscript[a, 36,10]-8 Subscript[a, 36,22]-7 Subscript[a, 37,24]-7 Subscript[a, 37,30]-6 Subscript[a, 42,22]-Subscript[a, 43,10]-Subscript[a, 43,14]-Subscript[a, 43,26]-2 Subscript[a, 44,17]-2 Subscript[a, 44,25]-Subscript[a, 45,18]-Subscript[a, 45,30]-Subscript[a, 47,13]-Subscript[a, 47,25]-Subscript[a, 49,22]>=0,Subscript[a, 49,22]>=0,-(227/42)+5 Subscript[a, 31,22]+5 Subscript[a, 36,22]+3 Subscript[a, 42,22]+Subscript[a, 49,22]>=0,-(489/56)+5 Subscript[a, 31,22]+8 Subscript[a, 36,22]+6 Subscript[a, 42,22]+Subscript[a, 49,22]>=0,-(137803198891/377814360)+41 Subscript[a, 1,24]+41 Subscript[a, 1,25]+38 Subscript[a, 4,5]+38 Subscript[a, 4,18]+38 Subscript[a, 5,4]+38 Subscript[a, 5,8]+38 Subscript[a, 5,17]+33 Subscript[a, 7,5]+33 Subscript[a, 7,13]+29 Subscript[a, 10,10]+29 Subscript[a, 10,24]+31 Subscript[a, 11,2]+31 Subscript[a, 11,4]+31 Subscript[a, 11,8]+30 Subscript[a, 12,14]+30 Subscript[a, 12,17]+33 Subscript[a, 13,2]+33 Subscript[a, 13,10]+24 Subscript[a, 17,2]+24 Subscript[a, 17,14]+21 Subscript[a, 19,4]+21 Subscript[a, 19,8]+21 Subscript[a, 19,18]+21 Subscript[a, 19,26]+16 Subscript[a, 23,2]+16 Subscript[a, 23,5]+18 Subscript[a, 24,4]+18 Subscript[a, 24,13]+18 Subscript[a, 24,26]+20 Subscript[a, 25,18]+20 Subscript[a, 25,26]+18 Subscript[a, 26,5]+18 Subscript[a, 26,17]+17 Subscript[a, 27,26]+17 Subscript[a, 27,30]+15 Subscript[a, 28,4]+15 Subscript[a, 28,13]+13 Subscript[a, 29,8]+13 Subscript[a, 29,10]+13 Subscript[a, 29,24]+12 Subscript[a, 30,24]+12 Subscript[a, 30,25]+9 Subscript[a, 31,13]+9 Subscript[a, 31,22]+9 Subscript[a, 31,26]+8 Subscript[a, 32,25]+8 Subscript[a, 32,26]+9 Subscript[a, 34,5]+9 Subscript[a, 34,14]+13 Subscript[a, 36,10]+13 Subscript[a, 36,22]+12 Subscript[a, 37,24]+12 Subscript[a, 37,30]+9 Subscript[a, 42,22]+3 Subscript[a, 43,10]+3 Subscript[a, 43,14]+3 Subscript[a, 43,26]+3 Subscript[a, 44,17]+3 Subscript[a, 44,25]+2 Subscript[a, 45,18]+2 Subscript[a, 45,30]+Subscript[a, 46,17]+Subscript[a, 47,13]+Subscript[a, 47,25]+3 Subscript[a, 49,22]>=0}

The variables are vars={Subscript[a, 1,24],Subscript[a, 1,25],Subscript[a, 4,5],Subscript[a, 4,18],Subscript[a, 5,4],Subscript[a, 5,8],Subscript[a, 5,17],Subscript[a, 7,5],Subscript[a, 7,13],Subscript[a, 10,10],Subscript[a, 10,24],Subscript[a, 11,2],Subscript[a, 11,4],Subscript[a, 11,8],Subscript[a, 12,14],Subscript[a, 12,17],Subscript[a, 13,2],Subscript[a, 13,10],Subscript[a, 17,2],Subscript[a, 17,14],Subscript[a, 19,4],Subscript[a, 19,8],Subscript[a, 19,18],Subscript[a, 19,26],Subscript[a, 23,2],Subscript[a, 23,5],Subscript[a, 24,4],Subscript[a, 24,13],Subscript[a, 24,26],Subscript[a, 25,18],Subscript[a, 25,26],Subscript[a, 26,5],Subscript[a, 26,17],Subscript[a, 27,26],Subscript[a, 27,30],Subscript[a, 28,4],Subscript[a, 28,13],Subscript[a, 29,8],Subscript[a, 29,10],Subscript[a, 29,24],Subscript[a, 30,24],Subscript[a, 30,25],Subscript[a, 31,13],Subscript[a, 31,22],Subscript[a, 31,26],Subscript[a, 32,25],Subscript[a, 32,26],Subscript[a, 34,5],Subscript[a, 34,14],Subscript[a, 36,10],Subscript[a, 36,22],Subscript[a, 37,24],Subscript[a, 37,30],Subscript[a, 42,22],Subscript[a, 43,10],Subscript[a, 43,14],Subscript[a, 43,26],Subscript[a, 44,17],Subscript[a, 44,25],Subscript[a, 45,18],Subscript[a, 45,30],Subscript[a, 46,17],Subscript[a, 47,13],Subscript[a, 47,25],Subscript[a, 49,22]}

Now I need to get a exact rational solution of p. A direct approach is FindInstance[p, vars, Rationals] // AbsoluteTiming, and it takes about 4 seconds. I wonder if there is a faster way to solve the problem, thanks.

Answer 1

Get the same result about four times faster with NMinimize and rationalization. (Using the format for variables, @BobHanlon proposed.)

pp = p /. Subscript[a, m_, n_] :> a[m, n];

vars2 = Union[Cases[pp, _a, Infinity]];

nmin = NMinimize[{1, pp}, vars2, 
    WorkingPrecision -> 200]; // AbsoluteTiming

(*   {0.8968761, Null}   *)

solnmin = vars2 /. nmin[[2]] // Rationalize[#, 0] &

pp /. (nmin[[2]] // Rationalize[#, 0] &)

(*   all True   *)

(fi = FindInstance[pp, vars2, Rationals];) // AbsoluteTiming

(*   {4.2813033, Null}   *)

solnmin == vars2 /. fi[[1]]

(*   True   *)