Return to Answer

This is a bit too long to be a comment. Let me phrase this probabilistically by saying that the coefficients $c_i$ are independent (discrete) random variables uniformly distributed on the finite set $$ \big\{\, -M,-(M-1),\dotsc, -1,0, 1, \dotsc, (M-1), M\,\big\}. $$ Denote by $N_d(f)$ the number of real zeros of the random degree $d$ polynomial $f$ whose coefficients $c_i$ are random variables as above. Your question asks for the probability distribution of the random variable $N_d(f)$. This is a very difficult question for fixed $d$. However, for fixed $M$ and $d\to\infty$ some nontrivial information is available.

A highly nontrivial result of Ibragimov and Maslova

The mean number of real zeros of random polynomials. I. Coefficients with zero mean. (Russian) Teor. Verojatnost. i Primenen. 16 1971 229–248.

generalizing earlier work of Kac, Erdos-Offord and D.C. Stevens shows that the expectation of $N_d(f)$ satisfies the asymptotic estimate $\newcommand{\bE}{\mathbb{E}}$

$$ \mu_d:=\bE\big[ N_d(f)\big]\sim \frac{2}{\pi}\log d\;\;\mbox{as $d\to\infty$}. $$

Moreover, the variance of this random variable is about the same size. A result of N. B. Maslova

The variance of the number of real roots of random polynomials, (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 36–51.

shows that $\DeclareMathOperator{\var}{var}$ the variance of $N_d(f)$ satisfies the asymptotic estimate $$ \sigma^2_d:=\var\big[ N_d(f)\big]\sim \frac{1}{\pi}\left(1-\frac{2}{\pi}\right)\log d\;\;\mbox{as $d\to\infty$}. $$

This suggests that the typical polynomial in your family does not have too many roots, compared to its degree. Finally, another result of N. B. Maslova

The distribution of the number of real roots of random polynomials, (Russian. English summary) Teor. Verojatnost. i Primenen. 19 (1974),488–500.

shows that the normalized random variable $$ Z_d=\frac{1}{\sigma_d}\Big( N_d(f)-\mu_d\Big) $$ converges in distribution to a standard normal random variable. You can use this result to estimate the probability that the number of zeros of $f$ lies in an interval of the form $$ [\mu_d+a\sigma_d, \mu_d+b\sigma_d], \;\; a, b\in\mathbb{R}, \;\; a<b. $$

This is a bit too long to be a comment. Let me phrase this probabilistically by saying that the coefficients $c_i$ are independent (discrete) random variables uniformly distributed on the finite set $$ \big\{\, -M,-(M-1),\dotsc, -1,0, 1, \dotsc, (M-1), M\,\big\}. $$ Denote by $N_d(f)$ the number of real zeros of the random degree $d$ polynomial $f$ whose coefficients $c_i$ are random variables as above. Your question asks for the probability distribution of the random variable $N_d(f)$. This is a very difficult question for fixed $d$. However, for fixed $M$ and $d\to\infty$ some nontrivial information is available.

A highly nontrivial result of Ibragimov and Maslova

The mean number of real zeros of random polynomials. I. Coefficients with zero mean. (Russian) Teor. Verojatnost. i Primenen. 16 1971 229–248.

generalizing earlier work of Kac, Erdos-Offord and D.C. Stevens shows that the expectation of $N_d(f)$ satisfies the asymptotic estimate $\newcommand{\bE}{\mathbb{E}}$

$$ \mu_d:=\bE\big[ N_d(f)\big]\sim \frac{2}{\pi}\log d\;\;\mbox{as $d\to\infty$}. $$

Recently (see this paper) this result was improved to $$ \mu_d:=\bE\big[ N_d(f)\big]= \frac{2}{\pi}\log d+O(1)\;\;\mbox{as $d\to\infty$}. $$

Remarkably, the variance of the random variable $N_d$ is about the same size. A result of N. B. Maslova

The variance of the number of real roots of random polynomials, (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 36–51.

shows that $\DeclareMathOperator{\var}{var}$ the variance of $N_d(f)$ satisfies the asymptotic estimate $$ \sigma^2_d:=\var\big[ N_d(f)\big]\sim \frac{1}{\pi}\left(1-\frac{2}{\pi}\right)\log d\;\;\mbox{as $d\to\infty$}. $$

This suggests that the typical polynomial in your family does not have too many roots, compared to its degree. Finally, another result of N. B. Maslova

The distribution of the number of real roots of random polynomials, (Russian. English summary) Teor. Verojatnost. i Primenen. 19 (1974),488–500.

shows that the normalized random variable $$ Z_d=\frac{1}{\sigma_d}\Big( N_d(f)-\mu_d\Big) $$ converges in distribution to a standard normal random variable. You can use this result to estimate the probability that the number of zeros of $f$ lies in an interval of the form $$ [\mu_d+a\sigma_d, \mu_d+b\sigma_d], \;\; a, b\in\mathbb{R}, \;\; a<b. $$

This is a bit too long to be a comment. Let me phrase this probabilistically by saying that the coefficients $c_i$ are independent (discrete) random variables uniformly distributed on the finite set $$ \big\{\, -M,-(M-1),\dotsc, -1,0, 1, \dotsc, (M-1), M\,\big\}. $$ Denote by $N_d(f)$ the number of real zeros of the random degree $d$ polynomial $f$ whose coefficients $c_i$ are random variables as above. Your question asks for the probability density of the random variable $N_d(f)$. This is a very difficult question for fixed $d$. However, for fixed $M$ and $d\to\infty$ some nontrivial information is available.

A highly nontrivial result of Ibragimov and Maslova

The mean number of real zeros of random polynomials. I. Coefficients with zero mean. (Russian) Teor. Verojatnost. i Primenen. 16 1971 229–248.

generalizing earlier work of Kac, Erdos-Offord and D.C. Stevens shows that the expectation of $N_d(f)$ satisfies the asymptotic estimate $\newcommand{\bE}{\mathbb{E}}$

$$ \mu_d:=\bE\big[ N_d(f)\big]\sim \frac{2}{\pi}\log d\;\;\mbox{as $d\to\infty$}. $$

Moreover, the variance of this random variable is about the same size. A result of N. B. Maslova

The variance of the number of real roots of random polynomials, (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 36–51.

shows that $\DeclareMathOperator{\var}{var}$ the variance of $N_d(f)$ satisfies the asymptotic estimate $$ \sigma^2_d:=\var\big[ N_d(f)\big]\sim \frac{1}{\pi}\left(1-\frac{2}{\pi}\right)\log d\;\;\mbox{as $d\to\infty$}. $$

This suggests that the typical polynomial in your family does not have too many roots, compared to its degree. Finally, another result of N. B. Maslova

The distribution of the number of real roots of random polynomials, (Russian. English summary) Teor. Verojatnost. i Primenen. 19 (1974),488–500.

shows that the normalized random variable $$ Z_d=\frac{1}{\sigma_d}\Big( N_d(f)-\mu_d\Big) $$ converges in distribution to a standard normal random variable. You can use this result to estimate the probability that the number of zeros of $f$ lies in an interval of the form $$ [\mu_d+a\sigma_d, \mu_d+b\sigma_d], \;\; a, b\in\mathbb{R}, \;\; a<b. $$

This is a bit too long to be a comment. Let me phrase this probabilistically by saying that the coefficients $c_i$ are independent (discrete) random variables uniformly distributed on the finite set $$ \big\{\, -M,-(M-1),\dotsc, -1,0, 1, \dotsc, (M-1), M\,\big\}. $$ Denote by $N_d(f)$ the number of real zeros of the random degree $d$ polynomial $f$ whose coefficients $c_i$ are random variables as above. Your question asks for the probability distribution of the random variable $N_d(f)$. This is a very difficult question for fixed $d$. However, for fixed $M$ and $d\to\infty$ some nontrivial information is available.

A highly nontrivial result of Ibragimov and Maslova

The mean number of real zeros of random polynomials. I. Coefficients with zero mean. (Russian) Teor. Verojatnost. i Primenen. 16 1971 229–248.

generalizing earlier work of Kac, Erdos-Offord and D.C. Stevens shows that the expectation of $N_d(f)$ satisfies the asymptotic estimate $\newcommand{\bE}{\mathbb{E}}$

$$ \mu_d:=\bE\big[ N_d(f)\big]\sim \frac{2}{\pi}\log d\;\;\mbox{as $d\to\infty$}. $$

Moreover, the variance of this random variable is about the same size. A result of N. B. Maslova

The variance of the number of real roots of random polynomials, (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 36–51.

shows that $\DeclareMathOperator{\var}{var}$ the variance of $N_d(f)$ satisfies the asymptotic estimate $$ \sigma^2_d:=\var\big[ N_d(f)\big]\sim \frac{1}{\pi}\left(1-\frac{2}{\pi}\right)\log d\;\;\mbox{as $d\to\infty$}. $$

This suggests that the typical polynomial in your family does not have too many roots, compared to its degree. Finally, another result of N. B. Maslova

The distribution of the number of real roots of random polynomials, (Russian. English summary) Teor. Verojatnost. i Primenen. 19 (1974),488–500.

shows that the normalized random variable $$ Z_d=\frac{1}{\sigma_d}\Big( N_d(f)-\mu_d\Big) $$ converges in distribution to a standard normal random variable. You can use this result to estimate the probability that the number of zeros of $f$ lies in an interval of the form $$ [\mu_d+a\sigma_d, \mu_d+b\sigma_d], \;\; a, b\in\mathbb{R}, \;\; a<b. $$

This is a bit too long to be a comment. Let me phrase this probabilistically. that the coefficients $c_i$ are independent (discrete) random variables uniformly distributed on the finite set $$ \big\{\, -M,-(M-1),\dotsc, -1,0, 1, \dotsc, (M-1), M\,\big\}. $$ Denote by $N_d(f)$ the number of real zeros of the random degree $d$ polynomial $f$ whose coefficients $c_i$ are random variables as above. Your question asks for the probability density of the random variable $Z_d(f)$. This is a very difficult question for fixed $d$. However, for fixed $M$ and $d\to\infty$ some nontrivial information is available.

A highly nontrivial result of Ibragimov and Maslova

The mean number of real zeros of random polynomials. I. Coefficients with zero mean. (Russian) Teor. Verojatnost. i Primenen. 16 1971 229–248.

generalizing earlier work of Erdos-Offord and D.C. Stevens shows that the expectation of $N_d(f)$ satisfies the asymptotic estimate $\newcommand{\bE}{\mathbb{E}}$

$$ \mu_d:=\bE\big[ N_d(f)\big]\sim \frac{2}{\pi}\log d\;\;\mbox{as $d\to\infty$}. $$

Moreover, the variance of this random variable is about the same size. A result of N. B. Maslova

The variance of the number of real roots of random polynomials, (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 36–51.

shows that $\DeclareMathOperator{\var}{var}$ the variance of $N_d(f)$ satisfies the asymptotic estimate $$ \sigma^2_d:=\var\big[ N_d(f)\big]\sim \frac{1}{\pi}\left(1-\frac{2}{\pi}\right)\log d\;\;\mbox{as $d\to\infty$}. $$

This suggests that the typical polynomial in your family does not have too many roots, compared to its degree. Finally, another result of N. B. Maslova

The distribution of the number of real roots of random polynomials, (Russian. English summary) Teor. Verojatnost. i Primenen. 19 (1974),488–500.

shows that the normalized random variable $$ Z_d=\frac{1}{\sigma_d}\Big( N_d(f)-\mu_d\Big) $$ converges in distribution to a standard normal random variable. You can use this result to estimate the probability that the number of zeros of $f$ lies in an interval of the form $$ [\mu_d+a\sigma_d, \mu_d+b\sigma_d], \;\; a, b\in\mathbb{R}, \;\; a<b. $$

This is a bit too long to be a comment. Let me phrase this probabilistically by saying that the coefficients $c_i$ are independent (discrete) random variables uniformly distributed on the finite set $$ \big\{\, -M,-(M-1),\dotsc, -1,0, 1, \dotsc, (M-1), M\,\big\}. $$ Denote by $N_d(f)$ the number of real zeros of the random degree $d$ polynomial $f$ whose coefficients $c_i$ are random variables as above. Your question asks for the probability density of the random variable $N_d(f)$. This is a very difficult question for fixed $d$. However, for fixed $M$ and $d\to\infty$ some nontrivial information is available.

A highly nontrivial result of Ibragimov and Maslova

The mean number of real zeros of random polynomials. I. Coefficients with zero mean. (Russian) Teor. Verojatnost. i Primenen. 16 1971 229–248.

generalizing earlier work of Kac, Erdos-Offord and D.C. Stevens shows that the expectation of $N_d(f)$ satisfies the asymptotic estimate $\newcommand{\bE}{\mathbb{E}}$

$$ \mu_d:=\bE\big[ N_d(f)\big]\sim \frac{2}{\pi}\log d\;\;\mbox{as $d\to\infty$}. $$

Moreover, the variance of this random variable is about the same size. A result of N. B. Maslova

The variance of the number of real roots of random polynomials, (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 36–51.

shows that $\DeclareMathOperator{\var}{var}$ the variance of $N_d(f)$ satisfies the asymptotic estimate $$ \sigma^2_d:=\var\big[ N_d(f)\big]\sim \frac{1}{\pi}\left(1-\frac{2}{\pi}\right)\log d\;\;\mbox{as $d\to\infty$}. $$

This suggests that the typical polynomial in your family does not have too many roots, compared to its degree. Finally, another result of N. B. Maslova

The distribution of the number of real roots of random polynomials, (Russian. English summary) Teor. Verojatnost. i Primenen. 19 (1974),488–500.

shows that the normalized random variable $$ Z_d=\frac{1}{\sigma_d}\Big( N_d(f)-\mu_d\Big) $$ converges in distribution to a standard normal random variable. You can use this result to estimate the probability that the number of zeros of $f$ lies in an interval of the form $$ [\mu_d+a\sigma_d, \mu_d+b\sigma_d], \;\; a, b\in\mathbb{R}, \;\; a<b. $$