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May 30th, 2012, 05:49 AM   #1
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Solvability of equation

Hi,

my computer science problem ends solving of equations and I need to know if there exists way how to solve it. I know it has integer solution.
The following system of equations is easy but I need to solve it for situations when the structure of problem is the same but parameters are from a0 to a20 and from b0 to b20. The only way I found is the substitution method, but it fails already for systems with a0 to a4 ... because it goes to very long polynomial equations.

a0*b0 = 31680
a0*b1+a1*b0 = -43008
a0*b2+a1*b1+a2*b0 = 22084
a0+a1*b2+a2*b1+b0=-5460
a1+a2*b2+b1 = 685
a2+b2 = -42

For this a bit more complex system I'm not able to find the solution:
a0*b0 = 18162144000
a0*b1 + a1*b0 = -37815724800
a0*b2 + a1*b1 + a2*b0 = 34351382160
a1*b2 + a0*b3 + a3*b0 + a2*b1 = -18067743072
a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 = 6147827896
-75*a0 + a1*b4 + a2*b3 + a3*b2 + a4*b1 - 27*b0 = -1429787040
a0 - 75*a1 + a2*b4 + a3 b3 + a4 b2 - 27 b1 + b0 = 233659985
a1 - 75*a2 + a3*b4 + a4*b3 - 27*b2 + b1 = -27102726
a2 - 75*a3 + a4*b4 - 27*b3 + b2 = 2219703
-75*a4 + a3 + b3 - 27*b4 = -125460
a4 + 2025 + b4 = 4655

Thanks!

Cyrrus
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May 31st, 2012, 06:01 PM   #2
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Re: Solvability of equation

There's a theorem that states that if a polynomial is irreducible in where is a prime number, then the polynomial is irreducible in the rational numbers. However, cannot divide the leading coefficient. In your case that's your greatest for . Consider something like and go from there. This helps to determine if whether or not the system of equations actually has a solution in the rational numbers. I'll see if I can help you any further, since you seem sure that it does have a solution.

Edit: I was able to use substitution to get to a point where solving it just involved a lot of computation. Substitution should be doable if you use a lot of short-hands and formulas. Here's are some pictures.
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June 4th, 2012, 12:21 AM   #3
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Re: Solvability of equation

Hi Xhin, thanks for your reply. As far I understand the problem I know it goes to 2 situations. The system has solution in R or it doesn't have. I need to distinguish between these two cases. If I find the solution the better but it's not neccessary.
I don't understand how to determine the solvability in Z2. Can you send me the scans with some description in better quality to email? I'm not able to read it and follow your idea. cyrrussmith I'm at gmail.com. Thanks again.
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