## A non-NP-complete algorithm for a quasi-fixed polynomial problem.(English)Zbl 1276.39012

Summary: Let $$F : \mathbb R \times \mathbb R \to \mathbb R$$ be a real-valued polynomial function of the form $$F(x, y) = \sum^s_{i = 0}f_i(x)y^i$$, with degree of $$y$$ in $$F(x, y) = s \geq 1$$, $$x \in \mathbb R$$. An irreducible real-valued polynomial function $$p(x)$$ and a nonnegative integer $$m$$ are given to find a polynomial function $$y(x) \in \mathbb R[x]$$ satisfying the following expression: $$F(x, y(x)) = cp^m(x)$$ for some constant $$c \in \mathbb R$$. The constant $$c$$ is dependent on the solution $$y(x)$$, namely, a quasi-fixed (polynomial) solution of the polynomial-like equation $$(\ast)$$. In this paper, we provide a non-NP-complete algorithm to solve all quasi-fixed solutions if the equation $$(\ast)$$ has only a finite number of quasi-fixed solutions.

### MSC:

 39B22 Functional equations for real functions 12D99 Real and complex fields 68Q25 Analysis of algorithms and problem complexity
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### References:

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