# Solving system of polynomial equations over rationals

I have to solve lots of systems of polynomial equations like the one bellow. There are 6 polynomials in 6 variables $$a1, a2, b1, b2, b3, b4$$. All polynomials should be equal to $$0$$. I need only rational solutions. I tried to solve it in Mathematica by Solve then tried to compute Groebner Basis both in Mathematica and in SageMath but neither is able to compute it. I let them to run about a day without any output.

What are other methods that would be efficient to solve such problems?

{-272853213601 + 114339252960*a2 - 4841413740*a2^2 + 296664007680*b2 - 25123011840*a2*b2 -
32592015360*b2^2 - 4907531205*b3 + 6155208630*b3^2 - 3860046090*b3^3 + 1210353435*b3^4 -
151807041*b3^5 + 312814245280*b4 - 97612876080*a2*b4 + 1518070410*a2^2*b4 -
253265840640*b2*b4 + 7877554560*a2*b2*b4 + 10219530240*b2^2*b4 - 146051082720*b4^2 +
29048482440*a2*b4^2 + 75369035520*b2*b4^2 + 35852138640*b4^3 - 3036140820*a2*b4^3 -
7877554560*b2*b4^3 - 4841413740*b4^4 + 303614082*b4^5,
-121828703201 - 1128406464*a1 + 303614082*a1^2 + 24547177584*a2 - 303614082*a2^2 -
2927757312*b1 + 1575510912*a1*b1 + 2043906048*b1^2 + 123022775808*b2 - 6600113280*a2*b2 -
15080712192*b2^2 + 1480577211*b3 + 146055798*b3^2 - 1347744906*b3^3 + 816475707*b3^4 -
151807041*b3^5 + 171636450272*b4 - 32717479104*a2*b4 + 303614082*a2^2*b4 -
135541762560*b2*b4 + 3151021824*a2*b2*b4 + 6131718144*b2^2*b4 - 95005523376*b4^2 +
13441077468*a2*b4^2 + 49947954048*b2*b4^2 + 26959925088*b4^3 - 1821684492*a2*b4^3 -
6302043648*b2*b4^3 - 4176745074*b4^4 + 303614082*b4^5,
-157794053 - 2365632*a1 - 20339456*a2 + 2907756*a2^2 - 10533888*b1 + 2365632*a1*b1 +
6137856*b1^2 + 131946240*b2 + 5178816*a2*b2 - 15925248*b2^2 + 219303*b3 - 2925810*b3^2 +
428238*b3^3 + 1269063*b3^4 - 455877*b3^5 + 273722192*b4 + 5853992*a2*b4 -
911754*a2^2*b4 - 188327424*b2*b4 + 9206784*b2^2*b4 - 178267752*b4^2 + 5470524*a2*b4^2 +
89190720*b2*b4^2 + 58666164*b4^3 - 1823508*a2*b4^3 - 14193792*b2*b4^3 - 10546776*b4^4 +
911754*b4^5, 1838593 + 3492*a1 + 24642*a1^2 + 946868*a2 - 24642*a2^2 + 293760*b1 +
63936*a1*b1 - 82944*b1^2 - 1109376*b2 - 267840*a2*b2 + 82944*b2^2 + 54693*b3 -
11574*b3^2 - 49590*b3^3 - 2331*b3^4 + 12321*b3^5 - 3856016*b4 - 1229144*a2*b4 +
24642*a2^2*b4 + 1900800*b2*b4 + 127872*a2*b2*b4 - 82944*b2^2*b4 + 2909904*b4^2 +
471528*a2*b4^2 - 1187136*b2*b4^2 - 1102172*b4^3 - 49284*a2*b4^3 + 255744*b2*b4^3 +
231102*b4^4 - 24642*b4^5, 1829 - 384*a1 + 2272*a2 + 236*a2^2 + 768*b1 + 384*a1*b1 -
768*b2 - 384*a2*b2 - 199*b3 - 206*b3^2 - 14*b3^3 + 89*b3^4 + 37*b3^5 - 5904*b4 -
3808*a2*b4 - 74*a2^2*b4 + 1536*b2*b4 + 384*a2*b2*b4 + 5168*b4^2 + 2304*a2*b4^2 -
1152*b2*b4^2 - 2216*b4^3 - 444*a2*b4^3 + 384*b2*b4^3 + 532*b4^4 - 74*b4^5,
-1 + 40*a1 + 10*a1^2 - 40*a2 - 10*a2^2 - 5*b3 - 10*b3^2 - 10*b3^3 - 5*b3^4 - b3^5 +
80*b4 + 80*a2*b4 + 10*a2^2*b4 - 80*b4^2 - 60*a2*b4^2 + 40*b4^3 + 20*a2*b4^3 - 10*b4^4 +
2*b4^5}

• Can you explain the background. Why do you only want rational solutions? Is there some physics or engineering behind this? Dec 2, 2019 at 9:04
• It is a pure mathematical problem of solving some Diophantine equations. There are lots of possibilities to check whether some underdetermined equation has rational solution. Each possibility yields to a system of equations (sometimes overdetermined). Then most of them have no rational solution and few of them have one or two rational solutions. Dec 2, 2019 at 9:26

$$a1 = -1008/1369, a2 = -11792/12321,b1 = 1, b2 = 1, b3 = 59/37, b4 = 118/37$$
• First step: Resultant[{eq_1,eq_2,eq_3,eq_4,eq_5},eq_6,a1]//Factor we get system {new_eq_1,new_eq_2,new_eq_3,new_eq_4,new_eq_5} without variable a1. Second step: Resultant[{new_eq_1,new_eq_2,new_eq_3,new_eq_4},new_eq_5,a2]//Factor we get factor (-118 + 37 b4). And etc. Dec 2, 2019 at 12:51
• Try Resultant[{new_eq_2,new_eq_3,new_eq_4,new_eq_5},new_eq_1,a2]//Factor. Second parametr of Resultant need try minimal polynomial from system. All integer factors need deleted, and also deleted power of polynomial. Dec 2, 2019 at 13:42