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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 $[-1,1]$.

Question 1: What is the probability that $$\{f(x):x\in \mathbb R\} \subset [0,\infty)? $$i.e. that $f$ only takes non-negative values. Equivalently these $f$ can be written as $f=g^2+h^2$ for some $g,h \in \mathbb R[x]$. I think this probability is well-defined but I cannot prove nor disprove it. For $d=2$ the following easy argument shows that the probability, if it exists, is strictly positive: consider the set of polynomials $$ \{f=c_2x^2+c_1x+c_0:0\leq c_1<1/2<c_0,c_2\leq 1 \}$$ that has positive density. Each such polynomial has minimum value $$ \frac{-c_1^2+4c_0c_2}{4c_2}\geq \frac{-1/4+1}{4}=\frac{3}{16}>0.$$

Question 2: Now let us focus on the polynomials that only take non-negative values. As $f$ ranges among them, can we give a bound for the probability of the event $m_f>z,$ where $$m_f:=\inf\{f(x): x\in \mathbb R, f(x)>0\}?$$ I am only interested in rough bounds and only when $z\to+\infty$.

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  • $\begingroup$ Why do you add $f(x) >0$ condition in the definition of $m_f$? $\endgroup$
    – Fedor Petrov
    Commented Sep 17, 2021 at 17:56
  • $\begingroup$ @FedorPetrov : I think the question is to estimate, for large $z>0$, the probability of the union of the following two events: (i) $\{\sup f\le0\}$ (on which we have $m_f=\infty$) and (ii) $\{\inf f>z\}$. $\endgroup$
    – Iosif Pinelis
    Commented Sep 17, 2021 at 18:18
  • $\begingroup$ My guess is that the OP did not intend to include the event $\{\sup f \le 0\}$. Let's wait for clarification. $\endgroup$
    – Yuval Peres
    Commented Sep 18, 2021 at 16:17
  • $\begingroup$ Indeed, I totally forgot that the polynomials may take only negative values. I will edit the question. $\endgroup$
    – Dr. Pi
    Commented Sep 18, 2021 at 18:58
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    $\begingroup$ @Dr.Pi : I think that your questions (in the current form) can be restated simply as follows. Question 1: What is $P(\inf f\ge0)$? Question 2: What are rough bounds on $P(\inf f>z)$ as $z\to\infty$? $\endgroup$
    – Iosif Pinelis
    Commented Sep 19, 2021 at 0:39

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