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Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that $$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$ Suppose that $P(x)> 0$ for all $x\in \mathbb{R}$. Prove that $$\forall x\in\mathbb{R};\phantom{;}P(x)\ge \frac{1}{2^{195} T^{64}}$$

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Source? ${}{}{}$ –  Gerry Myerson Jan 19 '13 at 4:32
    
False: $a_0 = a_1 = a_2 = a_3 = a_4 = 0.$ –  Joshua Ciappara Jan 19 '13 at 5:12
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This is not easy. The general case is treated here: mate.dm.uba.ar/~perrucci/On_the_minimum_simplex.pdf –  ivan Jan 19 '13 at 12:12
    
ivan// Thanks !! –  hunminpark Jan 19 '13 at 13:00

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