Solving polynomial simultaneous equations

[Guest]

we assume a,b,c,d are unknown variables,whose solution are either 0 or 1.So a power of a variable equals to itself(e.g,a^(n)=a).Would u please help me find a proper way to sovle the following simultaneous equations whose solutions are either 0 or 1?
-d-c+cd+bd+bc+ac-bcd-acd-abd-2abc+2abcd=0.....(1)
d-2bd-ad+abd-abc+b+bcd-1=0 .....(2)
-bd+bc+bcd-ab+abd-2abcd+acd-c=0 .....(3)
-a-bd-bc+2bcd+ad-2acd+ac=0 .....(4)
ps:Those equations are not Boolean Equations.They are just normal polynomial equations which share the same solution set with the Boolean ones.

[Guest]

The easies way is to check all combinations (0,0,0,0), (1, 0, 0, 0) ...
a=0, b=0, c=0, d=0 is not a solution because of this equation:
d-2bd-ad+abd-abc+b+bcd-1=0 .....(2)

[Guest]

thank u man.but would please tell me your method to sovle it?Is there any effective algerithm except the exhaustive search?
looking forward to ur reply.

[Guest]

So a = 0 or a = 1

Solve the system if a = 0 and then solve the system for a=1

if a = 0
we have:
-d-c+cd+bd+bc-bcd = 0
d-2bd+b+bcd-1=0
-bd+bc+bcd-c=0
-bd-bc+2bcd=0

[olivia]

Agreed, look at the cases for 0 and 1.

In the first case where they all equal 0, since all four equations equal zero, you can equate the two sets of equations and use process of elimination. Hopefully that made sense.