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### Solving system of linear Diophantine equations

A is a $m \times n$ integer matrix with $m>n$. Consider the set $S=${-1,0,1}.

I want to enumerate all possible $X \in S^n$ such that $AX=0$

I tried using smith_form(), but then i could not figure out how to force the solution to belong to elements of $S$;

### Solving system of linear Diophantine equations

A is a $m \times n$ integer matrix with $m>n$. Consider the set $S \subset Z$. For example suppose $S=${-1,0,1}.

I want to enumerate all possible $X \in S^n$ such that $AX=0$

I tried using smith_form(), but then i could not figure out how to force the solution to belong to elements of $S$;

 3 No.3 Revision slelievre 9868 ●7 ●104 ●196 http://carva.org/samue...

### Solving system of linear Diophantine equations

A is a $m \times n$ integer matrix with $m>n$. Consider the set $S \subset Z$. For example suppose $S=${-1,0,1}.

I want to enumerate all possible $X \in S^n$ such that $AX=0$$AX=0$.

I tried using smith_form(), but then i could not figure out how to force the solution to belong to elements of $S$;$S$.

EDIT. Thank you Dima for the solution. It works well. I am adding example code here.

lb = -1
ub = 1
A = matrix(ZZ, [(8,2,10,0,12,2,0), (4,6,9,1,14,5,2), (2,0,1,1,0,1,0), (2,1,3,0,4,0,1)])
eq1 = [(0,8,2,10,0,12,2,0), (0,4,6,9,1,14,5,2), (0,2,0,1,1,0,1,0), (0,2,1,3,0,4,0,1)]
ieq1 = [(-lb,1,0,0,0,0,0,0), (ub,-1,0,0,0,0,0,0),
(-lb,0,1,0,0,0,0,0), (ub,0,-1,0,0,0,0,0),
(-lb,0,0,1,0,0,0,0), (ub,0,0,-1,0,0,0,0),
(-lb,0,0,0,1,0,0,0), (ub,0,0,0,-1,0,0,0),
(-lb,0,0,0,0,1,0,0), (ub,0,0,0,0,-1,0,0),
(-lb,0,0,0,0,0,1,0), (ub,0,0,0,0,0,-1,0),
(-lb,0,0,0,0,0,0,1), (ub,0,0,0,0,0,0,-1)]
p = Polyhedron(eqns=eq1, ieqs=ieq1, base_ring=QQ)
p.integral_points()


((-1, -1, 1, 1, 0, 0, 0),
(0, -1, -1, 1, 1, 0, 0),
(0, -1, 0, -1, 0, 1,1),
(0, 0, 0, 0, 0, 0, 0),
(0, 1, 0, 1, 0, -1, -1),
(0, 1, 1, -1, -1, 0,0),
(1, 1, -1, -1, 0, 0, 0))


which is exactly what i needed. Thanks again.

 4 No.4 Revision slelievre 9868 ●7 ●104 ●196 http://carva.org/samue...

### Solving system of linear Diophantine equations

A is a I have an $m \times n$ integer matrix $A$ with $m>n$. Consider the $m>n$, and a set $S \subset Z$. For \mathbb{Z}$, for example suppose$S=${-1,0,1}.$S=\{-1,0,1\}$. I want to enumerate all possible$X \in S^n$such that$AX=0$. I tried using smith_form(), but then i could not figure out how to force the solution to belong to elements of$S\$.

EDIT. Thank you Dima for the solution. It works well. I am adding an example code here.here to illustrate your solution.

lb = -1
ub = 1
A = matrix(ZZ, [(8,2,10,0,12,2,0), (4,6,9,1,14,5,2), (2,0,1,1,0,1,0), (2,1,3,0,4,0,1)])
eq1 = [(0,8,2,10,0,12,2,0), (0,4,6,9,1,14,5,2), (0,2,0,1,1,0,1,0), (0,2,1,3,0,4,0,1)]
ieq1 = [(-lb,1,0,0,0,0,0,0), (ub,-1,0,0,0,0,0,0),
(-lb,0,1,0,0,0,0,0), (ub,0,-1,0,0,0,0,0),
(-lb,0,0,1,0,0,0,0), (ub,0,0,-1,0,0,0,0),
(-lb,0,0,0,1,0,0,0), (ub,0,0,0,-1,0,0,0),
(-lb,0,0,0,0,1,0,0), (ub,0,0,0,0,-1,0,0),
(-lb,0,0,0,0,0,1,0), (ub,0,0,0,0,0,-1,0),
(-lb,0,0,0,0,0,0,1), (ub,0,0,0,0,0,0,-1)]
p = Polyhedron(eqns=eq1, ieqs=ieq1, base_ring=QQ)
p.integral_points()


((-1, -1, 1, 1, 0, 0, 0),