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I am a mathematician and I am very new to theoretical computer science.

The definition of P/NP problem I found in wiki is that:

  • P is the set of decision problems solvable in polynomial time by a deterministic Turing machine.
  • NP is the set of decision problems solvable in polynomial time by a nondeterministic Turing machine.
  • NP is the set of decision problems verifiable in polynomial time by a deterministic Turing machine.

Consider the problem of solving a one-variable higher-order degree polynomial equation(e.g., find all $x$ satisfied with $x^5+c_4x^4+c_3x^3+c_2x^2+c_1x^1+c_0=0$ with given $c_i \in \mathbb{C}$). Mathematicians have proved that for degree $\geq 5$ the polynomial equation does not have algebraic (changed the word 'analytic' to 'algebraic', thank Emil Jeřábek) solutions (See Abel–Ruffini theorem).

Hence, by the definition of P/NP complexity, this problem should not be in P class (otherwise we can get the solution for degree $\geq 5$ equations in polynomial time). And this problem can be verified in polynomial time, does that mean this problem in NP class? Or not? And Why?

Also, I found a paper called "On phase retrieval of finite-length sequences using the initial time sample", Haralambos Sahinoglou and Sergio Cabrera, IEEE Transactions on Circuits and Systems, 1991. In Section 2 and Section 3 of this paper, the authors state that they can get all the zeros of any one-variable high-order polynomial $R(z)$ and then form the zeros to a NP-complete problem, as a result, they can prove their original problem is NP-complete. Why do they believe that finding any polynomial's zeros can be done?

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  • $\begingroup$ The fact that degree-$≥5$ polynomial equations have no implication whatsoever about the problem belonging or not to $\mathsf P$. One can complement Neal's answer: There are polynomial-time algorithms for polynomial root finding over many rings: $ℤ$, $ℚ$, number fields, $𝔽_q$, even some algebraic closures, etc. It is also the case over $ℝ$ or $ℂ$ once you properly define your problem (with approximate values). $\endgroup$
    – Bruno
    Jul 13, 2023 at 7:25
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    $\begingroup$ The reason for no implication is that Abel-Ruffini theorem states that there is no solutions in radicals but general algorithms have no reason to use only radicals for computing the roots. $\endgroup$
    – Bruno
    Jul 13, 2023 at 7:29
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    $\begingroup$ You are confusing two different meanings of the word: "solvability" of a polynomial equation by an expression in radicals has nothing to do with "solvability" of a decision problem by an efficient algorithm. $\endgroup$ Jul 13, 2023 at 7:45
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    $\begingroup$ Also, the Abel–Ruffini theorem says nothing about analytic solutions. All polynomials have analytic solutions. The theorem is about algebraic solutions by means of radicals. $\endgroup$ Jul 13, 2023 at 7:48
  • $\begingroup$ @Bruno Thanks for your comments. I understand that you mentioned the general algorithms for computing roots must not base on radicals, as a result, if there is a set of all (base) P-complexity mathematical operations (like radicals, add/minus, multiplication/division) defined in the theoretical computer science field? Also, if there is a formal definition of deterministic Turing machine process/algorithm and where can I find it? The descriptions on online websites (like Wiki) are hard to understand. $\endgroup$ Jul 17, 2023 at 18:42

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Finding the factorization of a single-variable polynomial with rational coefficients is apparently solvable in poly time, as shown in [1]:

[1] Lenstra, A. K.; Lenstra, H. W., Jr.; Lovász, L. Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), no. 4, 515–534. (link).

The following posts discuss this:

This presumably gives you a polynomial-time algorithm to find all the real roots. Are you also interested in finding non-real (complex) roots?

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    $\begingroup$ I feel sorry that my question is not clear. My original question is that, what is the complexity of finding all "complex" roots of a single-variable polynomial with "complex" coefficients. Also, I want to ask the complexity of finding all "complex" roots of a single-variable polynomial with "real" coefficients. Thanks for your answer to the complexity of "real" roots with "rational" coefficients $\endgroup$ Jul 17, 2023 at 18:04

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