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How does one check for the positivity of coefficients of a rational function,say, for example $\frac{p_1(x,t)}{(1-xt)(1-x^2t)(1-x^3t)}$ where $p_1(x,t) = 1 + tx + 2t^2x^2 - 3x^3t^2 -x^5t^3 - x^{10}t^4$? In general, let $p_1(x,t)/p_2(x,t)$ be a rational function over field $k$,suppose the power series expansion of $1/p_2(x,t)$ has non-negative coefficients. If $p_1(x,t)$ is a polynomial with some negative coefficients, when can we say that the expansion of $p_1(x,t)/p_2(x,t)$ has non-negative coefficients?

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This seems to be a hard question. See this paper by Laffey et al (2012) for extensive references.

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