Solve system of linear Diophantine equations with conditions

Questions
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1

Hi everyone,
I would like to ask your help in the following problem, I want to find nonnegative solutions for a system of linear Diophantine equations using some conditions on the constant terms. For example, in the following linear system the variables are x,y_1,y_2,z_1,z_2,w, and n,a,b,c_1,c_2 are nonnegative integers:

x+w =a
y_1+y_2+nw=b
y_1+w-z_1=c_1
y_2+w-z_2=c_2

and we have the next conditions for the constants:

0\leq b-an, \;\; 0\leq a-c_1, \;\; 0\leq a-c_2, \;\; \mbox{and} \;\; 0\leq b-c_1-c_2.

I tried to use FindInstance[ ] but I had mistakes even if I don’t try to write the conditions for the integers n,a,b,c_1,c_2:

FindInstance[ x + w == a && y1 + y2 + n w == b && y1 + w - z1 == c1 && y2 + w - z2 == c2 && 0 <= x && 0 <= y1 && 0 <= y2 && 0 <= w &&
0 <= z1 && 0 <= z2, {x, y1, y2, w, z1, z2}, Integers]

FindInstance::exvar: “The system contains a nonconstant expression a independent of variables {x,y1,y2,w,z1,z2}”

If I try to add the conditions for the constants, then I obtain the same mistake.

Thanks.

2

I think it’s complaining about b, c1 and c2, which are not variables or numbers. Since your equations are all linear, try Solve or Reduce.

3

Thank you for your comment. I obtain the following using Solve: If I try to write the conditions on the constant I obtain the same mistake as in FindInstance. Now, if I don’t write the conditions for the constants, then I obtain a solution but some of the constants n,a,b,c_1, c_2 has assigned values. However, using SetAttributes[{n,a,b,c1,c2}, Constant] and only writting the conditions for the constants, the mistake was fixed and I obtain conditional expressions in terms of n,a,b,c_1,c_2.

Edit: Solve works writing only the conditions for the constants, and in the solution some of them has assigned values in terms of other constants.

4

That’s interesting. Could you attach your notebook? I’d like to see exactly how that worked.

5

Sure, I attached the notebook. I’m sorry, I got lost among all the calculations and attempts that I did: Solve works if I write the conditions for the constants, and in the solution some of them has assigned values in terms of other constants.

Now, I am trying to obtain a solution using the conditions for the constants and the condition that the solution is nonnegative.

6

Hi. I think you are missing some constraints.

const = Thread[ {a, b, n, c1, c2, b - a*n, a - c1, a - c2, b - c1 - c2} >= 0 ]
7

Thank you for your comment, I didn’t know Thread. That was a problem with FindInstance, such constraints for the constants made the mistake. I just try with Thread before FindInstance and the mistake remains, but before Solve and considering only the linear system everything works fine. Indeed, I didn’t use SetAttributes and requiring integer solutions, in the conditional expression obtained appears the hypothesis that a,b,c_1,c_2,n are integer numbers.

8

Seems to me you are trying to use FindInstance to find a symbolic instance. I don’t think it does that.