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Let the problem of the diophantic equation 0/1 be as follows.

Input : A polynomial equation on n variables whose coefficients are integers (ex : $2x^3_1 x_2 + x_1x^3_3 - 3x_4 = 8$)

Question: Does this equation have a solution in space {0,1}$^n$ ? (can we satisfy the equation by choosing for each variable the value 0 or 1?)

1 - How to prove that this diophantine equation 0/1 is NP-complete?

2 - How to prove that this diophantine equation 0/1 is Strongly NP-complete (ie its restriction to the case where all the coefficients are bounded by a polynomial of the number of variables is already NP-complete)

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    $\begingroup$ What does it mean for a diophantine equation to be NP-complete? NP-completeness is a category of decision problems. What decision problem do you have in mind? $\endgroup$ – Yuval Filmus Mar 13 at 18:16
  • $\begingroup$ Thanks for your comment, NP complete is good for me! but this is a strongly NP complete. How demonstrated that this problem is strongly NP complete. $\endgroup$ – tala Mar 14 at 13:24
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    $\begingroup$ As @YuvalFilmus said, It makes no sense to say that "this diophantine equation is strongly NP-complete" because 1) you did not specify what decision problem you are interested in, and 2) (strong) NP-completeness only makes sense over classes of instances. The answer to any single (fixed) instance of any problem in NP can be found in constant time. $\endgroup$ – Steven Mar 15 at 0:18
  • $\begingroup$ Okay, it's true what you said. I formulate the problem: Input: A polynomial equation on n variables whose coefficients are integers (example: $2x^3_1 x_2 + x_1x^3_3 - 3x_4 = 8$) Question: Does this equation have a solution in space {0,1}$^n$ (Can we satisfy the equation by choosing for each variable the value 0 or 1) How to prove that this diophantine equation 0/1 is strongly NP-complete? $\endgroup$ – tala Mar 15 at 1:14
  • $\begingroup$ @tala again, this instance has a fixed answer (we might not know it, but that is besides the point). Nothing to "compute", a simple look up. "Complexity" refers to the resources needed to compute answers to problems with infinite instances (if finite, a boring lookup is all what is needed). $\endgroup$ – vonbrand Mar 19 at 2:57
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This problem is strongly NP-complete; it is straightforward to reduce 3SAT to this problem.

The clause $C = (x_1 \lor \neg x_2 \lor x_3)$ can be encoded as the polynomial $p(x_1,x_2,x_3,\dots) = [(1-x_1)x_2(1-x_3)]^2$. The conjunction of clauses $\varphi = C_1 \land \dots \land C_m$ can be encoded as $q(x_1,\dots,x_n) = p_1(x_1,\dots,x_n) \cdots p_m(x_1,\dots,x_n)$, where $p_i(x_1,\dots,x_n)$ is the encoding of clause $C_i$. Then, the polynomial $q(x_1,\dots,x_n)$ has a solution in $\{0,1\}^n$ if and only if the original formula $\varphi$ is satisfiable.

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  • $\begingroup$ Thank you for your response, the 3-SAT reduction is also the idea I had before. Thanks for this example. Here therefore you have demonstrated question 2 that this is a problem is strongly NP-complete. Can we still use this same approach for question 1 which asks to demonstrate that the problem is NP-complete. I would also like to know the difference between NP-complete and strongly NP-complete in this case. $\endgroup$ – tala Mar 19 at 20:55
  • $\begingroup$ @tala, I suggest reading about the definition of NP-completeness and strongly NP-complete. You might need to ask a different question to learn about those concepts. Once you do, you should be able to answer that question yourself. $\endgroup$ – D.W. Mar 19 at 21:29

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