Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]$. Is there a bound in the literature for the probability of the event $m_f>z,$ where $$m_f:=\inf\{f(x): x\in \mathbb R, f(x)>0\}?$$ Especially when $z\to+\infty$. I know there are many results in this setting regarding the total number of real roots but I cannot find something about $m_f$.